2.2. Spatial Filtering Operation

So, in LES the larger eddies are computed with a time dependent simulation where the influence of this small eddies are incorporated through turbulence model.

In Large Eddy Simulation (LES), we focus on calculating the larger vortices or eddies present in a fluid flow. These larger structures are solved using a time-dependent simulation approach. It means that we track how these eddies change over time. However, the smaller eddies, which are too tiny to be computed directly due to their high number and complexity, are not actually solved for; instead, their effects are taken into account using a model that simulates their influence on the larger eddies.

Think of it like a weather report where forecasters predict large weather patterns (like storms), but they do not attempt to track every tiny cloud. Instead, they use models that estimate how smaller weather phenomena affect the larger storm's behavior.

The scales that are directly solved for on the grid are called the grid scales for large eddies. They are used for the large eddies and for the smaller one the Subgrid Scales (SGS).

In LES, we categorize scales of motion into two types: Grid Scale (GS) and Subgrid Scale (SGS). Grid Scale refers to the larger would-be captured eddies that are resolved directly on our computational grid. In contrast, Subgrid Scale refers to the smaller eddies that cannot be directly resolved and are instead modeled through other calculations. The resolution of the grid must be sufficiently smaller than the smallest eddy to accurately represent the fluid dynamics.

Imagine a video game where the main characters are visible and animated with high detail (the grid scale), but the background details are more blurred or simplified (the subgrid scale). The game designers make decisions on how to represent detail without needing to show every blade of grass or leaf.

Now, the filtering operation that we have talked about is defined by a filter function g of x, x dash, delta.

In LES, we utilize a filtering operation to distinguish between the larger and smaller eddies. This operation uses a mathematical function called a filter function, which acts to remove the smaller scales of motion from the calculation. This is essential for simplifying the model, allowing us to focus computational resources on the important larger scales while approximating the effects of smaller scales. The characteristics of the filter function are determined by parameters like the grid size.

Think of this filtering process like using a sieve to separate larger pasta shapes from smaller bits of semolina. The sieve allows the desired pasta shapes to remain while filtering out the finer particles.

So, now coming to the governing equations of LES the filtered momentum equation using the grid scale variables can be written.

In LES, we derive a filtered momentum equation using variables associated with the grid scale. This equation includes terms for their interactions and effects from the smaller scales, expressed through something called subgrid scale stress (SGS stress). This allows for modeling the interactions between resolved larger scales and unresolved smaller scales comprehensively.

Consider this as writing down the rules for a sports team. You note the main strategies (the large eddy equations) everyone should follow but also include some general advice on how to react to unforeseen circumstances (the subgrid scale modeling) that can arise during the game.

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