Free Surface Mesh Refinement

For more information on Adaptive Mesh Refinement, see Adaptive Mesh Refinement.

To ensure that a sharp interface is maintained, the Free Surface Mesh Refinement criterion avoids refining cells that already contain the interface. The algorithm attempts to identify the cells downstream (and, optionally, upstream) of the current interface position and mark them for refinement, to ensure that the interface resolution is maintained when the interface moves into these cells. However, when the Adaptive Mesh Interpolation option is set to Sharp Reconstruction, the Free Surface Mesh Refinement criterion also marks interface cells for refinement.

The algorithm is outlined below:

1. Identify the cells that contain the interface. The identification of the appropriate cells is based on the magnitude of the volume fraction gradient. A cell is recognized and marked as an interface cell when:

   $$\left|{\left(\nabla \alpha \right)}_i\right|>\frac{{\left(\nabla \alpha \right)}_{i,\max }}{r_i}$$

   (2600)

   for at least one component $i$ of the volume fraction gradient.

   ${\left(\nabla \alpha \right)}_{\max }$ denotes the maximum volume fraction gradient that is achievable on the given cell, and $r_i$ is a user-specified sensitivity parameter (the Resolution Criterion for Interface Detection property in the Free Surface Mesh Refinement criteria).

   The marked cells define the initial interface indicator $I_0$ such that:

   $$I_0=\{\begin{array}{c}\text{0 if the cell is not an interface cell}\\ \text{1 if the cell is an interface cell }\end{array}$$

   (2601)

2. The interface indicator $I_0$ is advected with the flow field to predict the interface position at the next AMR event. In some cases, it can be useful to also refine the mesh behind (upstream of) the interface. In these cases, the advection also takes place in the opposite flow direction.

   The time of the next AMR event is determined from the time-step size, the AMR period (specified as the Update Frequency of the Adaptive Mesh solver) and two user-specified parameters (set as the Swept Distance Estimation Factor [Upstream, Downstream] property in the Free Surface Mesh Refinement criteria):

   $$\begin{array}{c}{T}^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}}=t+C^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}}\cdot \mathrm{\Delta }t\cdot N_{\mathrm{A}\mathrm{M}\mathrm{R}}\\ T^{\mathrm{U}\mathrm{p}}=t+C^{\mathrm{U}\mathrm{p}}\cdot \mathrm{\Delta }t\cdot N_{\mathrm{A}\mathrm{M}\mathrm{R}}\end{array}$$

   (2602)

   where

   • $\mathrm{\Delta }t$ is the time step size

   • $N_{\mathrm{A}\mathrm{M}\mathrm{R}}$ is the AMR period

   • $C^{\mathrm{U}\mathrm{p}}$ , $C^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}}$ are the swept distance estimation factors, upstream and downstream respectively.

   An explicit Euler scheme is then used to solve two transport equations for the downstream and upstream markers:

   $${\partial }_t{I}^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{/}\mathrm{U}\mathrm{p}}+{\overrightarrow{u}}^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{/}\mathrm{U}\mathrm{p}}\cdot \nabla I^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{/}\mathrm{U}\mathrm{p}}=0\text{ for }t\in (t,t^{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{}\mathrm{A}\mathrm{M}\mathrm{R}})$$

   (2603)

   $$\begin{array}{c}{I}^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{/}\mathrm{U}\mathrm{p}}(t,\overrightarrow{x})=I_0\left(\overrightarrow{x}\right)\\ I^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{/}\mathrm{U}\mathrm{p}}(s,\overrightarrow{x})=1\end{array}$$

   (2604)

   for all $s$ and $\overrightarrow{x}$ where $I_0\left(\overrightarrow{x}\right)=1$ .

   Eqn. (2604) acts as a source term for the initial marker such that all of the cells between the initial and final interface position are marked for refinement during the transport process.

   A sub-stepping and rollback scheme is used to stabilize the explicit transport for large CFL numbers. This scheme starts with an initial time-step that corresponds to the global time-step and solves Eqn. (2603). If the transported marker field contains non-physical values (that is, it exceeds certain thresholds), the transport is repeated using smaller time-step sizes until the solution becomes stable.

   The transported marker $I$ containing both directions is determined as $I=\max (I^{\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}},I^{\mathrm{U}\mathrm{p}})$ .

3. The marker $I$ is extended by one additional safety cell layer.

4. Finally, the cells are marked for Adaptive Mesh Refinement according to the following rules:

   • if $e\in I_0$ , which represents all of the cells at the current interface position, and if Adaptive Mesh Interpolation is set to Injection, mark as keep. These cells are not modified at the next AMR event.

   • if $e\in I_0$ and Adaptive Mesh Interpolation is set to Sharp Reconstruction, mark as refine. The cells are then refined at the next AMR event.

   • if $e\in I_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}$ and $e\notin I_0$ , mark as refine. This represents all of the cells between the current and predicted interface positions. These cells are refined at the next AMR event.

   • else, mark as coarsen. These cells are coarsened at the next AMR event, or left unchanged if they are already at their original size.

   When the Limit Cell Size property in the Free Surface Mesh Refinement criteria is enabled, cells that were marked as refine above are marked as keep if refinement would result in child cells smaller than the specified Min Adaption Cell Size profile.

   Solution Initialization Procedure

   During solution initialization, the Free Surface Mesh Refinement criterion follows the same algorithmic steps as outlined above, but unconditionally marks all cells at the interface $e\in I_0$ as refine. The number of AMR events during this stage is determined by the maximum allowable refinement levels for all the criteria.

   After each sweep, the solution is re-initialized on the refined mesh, and the procedure is repeated. Because the re-initialized solution can be significantly different on the refined mesh (for example, by including flow or free surface features not resolvable on the preceding coarse mesh), the refinement can be non-optimal after each step. As such, the safety mechanism applied in Step 3 can result in one or two unnecessary cell layers, which are removed at the next AMR event. This situation can be avoided by increasing the number for the Max Refinement Level to allow for additional initialization sweeps, and by adjusting the Min Adaption Cell Size property to stop refinement at the desired resolution.