Arbitrary Lagrangian Euerian Method

In contrast to the IB method, the Arbitrary Lagrangian Eulerian (ALE) approach does not include the packed bed in the computational domain and instead uses a Dynamic Mesh to track the changing topography of the riverbed boundary. Figure 3-2 describes the ALE method whereby each node on the bottom fluid domain boundary is updated during each time step in the same way that the riverbed deformed due to erosion and deposition. The vertical displacement of an arbitrary node at the bed is determined by

where the set 𝐹𝐹 contains all the node’s neighbouring faces and ‖𝐹𝐹‖ denotes the number of neighbour faces. Note that decoupling by 𝑆𝑆𝑡𝑡 is only applicable if

Figure 3-2: Arbitrary Lagrangian Eulerian Method (Schneiderbauer & Pirker, 2014) 3.4.2.1 Smoothing of the Internal Mesh

When the dynamic mesh approach is used, the quality of the mesh can decrease significantly and cause numerical instabilities. The displaced nodes on the bed can overstretch the cells, increase the cell skewness and even cause negative cell volumes to occur (as demonstrated by Figure 3-3). Furthermore, the deformations may cause the law-

𝑅𝑅𝑚𝑚𝑚𝑚 = 1.098 𝑘𝑘 = 5.1 𝜏𝜏𝑤𝑤⁄ , 𝜌𝜌 (3-46) 𝑅𝑅𝑦𝑦𝑦𝑦= 0.655 𝑘𝑘 = 2.3 𝜏𝜏𝑤𝑤⁄ , 𝜌𝜌 (3-47) 𝑅𝑅𝑧𝑧𝑧𝑧= 0.247 𝑘𝑘 = 𝜏𝜏𝑤𝑤⁄ , 𝜌𝜌 (3-48) 𝑅𝑅𝑚𝑚𝑦𝑦= 𝑅𝑅𝑦𝑦𝑧𝑧=𝑅𝑅𝑚𝑚𝑧𝑧= −0.255 𝑘𝑘 = −𝜏𝜏𝑤𝑤⁄ . 𝜌𝜌 (3-49) 𝑆𝑆𝑅𝑅𝑆𝑆𝑅𝑅= 𝑋𝑋 𝑅𝑅𝑋𝑋− 𝑋𝑋 𝑅𝑅𝑖𝑖𝑖𝑖 , (3-50) ∆ℎ =_𝜌𝜌𝑆𝑆𝑡𝑡 ∆𝑡𝑡 𝑏𝑏 ‖𝐹𝐹‖ � 𝑉𝑉𝑠𝑠 𝐴𝐴𝑠𝑠𝑆𝑆𝑠𝑠𝑑𝑑,𝑠𝑠 𝑠𝑠∈𝐹𝐹 , _(3-51) 𝑣𝑣𝐼𝐼𝑖𝑖𝑚𝑚 ≫ ∆ℎ𝐼𝐼𝑎𝑎𝑚𝑚 𝑆𝑆𝑡𝑡 /∆𝑡𝑡 . (3-52)

of-the-wall distance 𝑌𝑌+ _{condition necessary for the formation of a boundary layer or} horseshoe vortex to be violated. In order to absorb the movement of the bed deformation, the interior mesh needs to be distorted along with the surface of the packed bed. Smoothing methods can be applied to adjust the interior nodes of the mesh while maintaining the number of nodes and their connectivity. In this instance, either the linearly-elastic-solid- based smoothing or boundary-layer-based smoothing methods are recommended. These methods are more computationally expensive but are better at preserving the mesh quality and required wall distance compared to other simpler methods such as the Laplacian smoothing method.

Figure 3-3: Example of mesh distortion within first cell layer adjacent to the riverbed boundary

With linearly-elastic-solid based smoothing, the motion of the interior mesh is treated as an elastic solid subjected to a vertical mesh displacement of ∆ℎ (or mesh displacement vector 𝐡𝐡). The motion of the interior mesh is governed by the following equations

where 𝛔𝛔(𝐡𝐡) is the stress tensor and 𝛆𝛆(𝐡𝐡) is the strain tensor of the mesh displacement vector 𝐡𝐡, 𝜇𝜇 is the shear modulus, λ _{is Lame’s first parameter and ύ is Poisson’s ratio taken as a} default of 0.45.

With boundary-layer-based smoothing, the nodes of each cell in the boundary layer are given the same displacement as that of the bed boundary, in order to preserve the quality and height of the boundary layer cells adjacent to the deformed boundary. Spring-based smoothing is then applied to the rest of the interior mesh using Hooke’s Law and a stiffness or spring constant factor 𝑘𝑘𝑐𝑐. The displacement ∆ℎ𝑖𝑖 for node 𝑖𝑖 with 𝑛𝑛 number of neighbouring cells is solved iteratively from the following equation using a Jacobi sweep

∇ ∙ 𝛔𝛔(𝐡𝐡) = 0 , (3-53)

𝛔𝛔(𝐡𝐡) =λ�𝑡𝑡𝐹𝐹𝛆𝛆(𝐡𝐡)�𝐈𝐈 + 2𝜇𝜇𝛆𝛆(𝐡𝐡) , (3-54)

𝛆𝛆(𝐡𝐡) =1₂(𝛁𝛁𝐡𝐡 + (𝛁𝛁𝐡𝐡)𝑇𝑇_{) , (3-55)}

where a value of 𝑘𝑘𝑐𝑐 = 0 would indicate that there is no damping on the motion of interior nodes.

3.4.2.2 Smoothing of the Bed Boundary

The ALE method deforms the riverbed boundary of the computational domain which can also cause a poor mesh quality to form. If neighbouring nodes undergo vastly different vertical displacement relative to a fine mesh resolution in one timestep, extreme irrecoverable irregularities such as those in Figure 3-4 can occur and cause numerical model instabilities, particularly at the start of a new simulation. The mesh motion resulting in these irregularities could be attributed to different rates of erosion and deposition due to the probabilistic or random nature of the sediment transport equations subjected to the turbulent effects of a fluctuating horseshoe vortex.

The proposed numerical model’s code was modified such that the user-defined function (UDF) for grid motion would loop over a face-zone instead of a cell-zone and to allow artificial smoothing to be applied to the face-zone or boundary.

Figure 3-4: Irregularities in the displacement of the cell nodes on the packed bed boundary at a bridge pier Artificial smoothing is performed after the initial movement of the nodes (before the hydrodynamic calculations) only if a node on the bed boundary is deformed more than a prescribed limit relative to its neighbouring nodes. If the difference between the vertical displacement of a given node i and the mean displacement of its neighbouring nodes ∆ℎ����_𝑖𝑖 exceeds a percent tolerance limit Ψ, at worst it will be replaced by the mean value. The equations are � 𝑘𝑘𝑖𝑖𝑖𝑖�∆ℎ𝑖𝑖− ∆ℎ𝑖𝑖� 𝑚𝑚 𝑖𝑖 = 0 , (3-57) 𝑘𝑘𝑖𝑖𝑖𝑖= 𝑘𝑘𝑐𝑐/��𝐡𝐡𝑖𝑖− 𝐡𝐡𝑖𝑖� , (3-58) ∆ℎ𝑖𝑖= �∆ℎ𝑖𝑖, �∆ℎ𝑖𝑖− ∆ℎ������ ≤𝚤𝚤 Ψ ∆ℎ����� 𝚤𝚤 ∆ℎ𝚤𝚤 �����, �∆ℎ𝑖𝑖− ∆ℎ������ >𝚤𝚤 Ψ ∆ℎ����� , 𝚤𝚤 (3-59) ∆ℎ ����_𝑖𝑖=1 𝑛𝑛 � ∆ℎ𝑖𝑖 𝑚𝑚 𝑖𝑖 , (3-60)

where the proposed tolerance limit of 0.1% controls the degree of natural erosion permitted per time-step. Enforcing an artificially smooth profile for numerical stability may reduce the solution accuracy. Therefore, a new variable Σ𝐻𝐻𝑖𝑖 that sums the difference �∆ℎ𝑖𝑖− ∆ℎ𝚤𝚤������ lost to smoothing with time at each node is defined and can be used to monitor the location and quantity of smoothing required.