Wall Modelling for LBM

What is a Boundary layer?

A boundary layer is a gradient in velocity that runs parallel to a physical wall.

Figure 1: Representation of a boundary layer and the velocity profile

Boundary layers can vary in thickness ranging from hundreds of meters (such as atmospheric boundary layers of land or sea) to hundredths of a millimeter (such as in supersonic flows). This makes it very tricky when it comes to resolving a boundary layer in CFD and, generally speaking, these layers are modelled to reduce the computational requirement of dealing with boundary layers.

How does thickness affect cost in CFD?

In CFD, to resolve a boundary layer we would need a very fine mesh all the way to the wall where the CFD code could solve the smallest scales in the viscous sublayer. However, the fineness of the mesh will depend upon the layer thickness and therefore, a higher Reynolds number will require more cells to solve the small scales. 

Figure 2: A representation of the mesh next to the velocity profile, illustrating the first cell distance.

The above figure shows how a layer relates to the mesh, where the thinner the layer, the finer the mesh requirement, and therefore the larger the mesh in general, this, of course, means more expense to the user to solve the problem accurately. Fortunately, we can model the boundary layer to reduce the cost. 

Wall Modelling

Wall modelling approaches are very different across different turbulence models, however, SimScale’s LBM solver mainly uses the K-omega SST model, and for the implementation, the Y+ is not required to have a strict first cell Y+ due to the approach it takes to wall modelling, however, it does perform best where Y+ is below 2000 for a coarsely refined model or 200 for a finely refined one.

Figure 3: the boundary layer, and the different regions that exist in a typical boundary layer, illustrated in dimensionless units.

Here we can see the different regions of the boundary layer and why when modelling the layer it’s encouraged to avoid the Log Law Region. However, K-omega SST models the layer up until the first cell then solves from there on, and this model has been proven to be very accurate in many industries including the aerospace industry. 

Although this model is highly accurate, more accurate models exist, namely LES or Large Eddy Simulation. LES is more accurate as thismodels only eddies smaller than the grid filter and solves the flow regime larger than the grid filter size. Amongst one of its downfalls is its inability in its standard form to model walls, therefore requiring a very fine mesh, or simply dealing with flows where wall interactions are least predominant. However, if a wall model were to be added, we could obtain the accuracy improvements without the requirement of such a fine mesh, and this is where the advantage of DES or Detached Eddy Simulation comes from. 

In the LBM solver two detached eddy models are available, the K-omega SST DDES (Delayed Detached Eddy Simulation) and the K-omega SST IDDES (Improved Delayed Detached Eddy Simulation). Here, the same wall requirements exist for the wall modelling however, at some point the near wall region transitions from K-omega SST to LES.

Figure 4: An illustration of where K-omega SST operates in the boundary layer for the different DES models, and where the large eddy model transitions within the boundary layer.

The difference however between DDES and IDDES is that IDDES blends from uRANS to LES in the buffer region which can be approximated to be somewhere between 5 < y+ < 30, whereas the DDES model blends from uRANS to LES in the log-law region 30 < Y+, therefore depending upon the Y+ values of your simulation might make a difference to which model you select. For example if your Y+ is around 100, then the DDES model would be better than IDDES, however, if the Y+ is below 5, the IDDES would be more suited.

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