1.1. Comparison with DNS

Another such technique is called Large Eddy simulation. See in the DNS one important thing to note was that we had the best accuracy but a lot of computational time is required. LES is sort of a tradeoff between the Reynolds average. In Reynolds average, we do many approximations, so the results are not that accurate compared to DNS, but LES is something which is a tradeoff between DNS and Reynolds average Navier-Stokes equation.

Large Eddy Simulation (LES) is a technique used in fluid dynamics to simulate turbulence. It offers a compromise between two other methods: Direct Numerical Simulation (DNS) and Reynolds-Averaged Navier-Stokes (RANS) equations. While DNS provides accurate results, it is computationally expensive, especially for turbulent flows. On the other hand, RANS simplifies the calculations by averaging over time but sacrifices accuracy. LES sits in between these two, capturing the essential large-scale features of turbulence while modeling the smaller scales. This allows for a more efficient simulation without losing too much accuracy.

Think of LES as a movie that highlights key plot points while summarizing parts that are less pivotal. Just like a movie shortens certain scenes to keep the essence intact while reducing length, LES captures the most important turbulent behaviors while simplifying smaller-scale details that don’t significantly affect the overall understanding.

So there is a big difference in the behaviors of large and small eddies in a turbulent flow field. We were talking in DNS about the length scales; we said that there will be vortices or a LES that are as big as the length of the flow. There will be LES at the time of dissipation if heat is very, very small, let us say order of 10 to the power -5 -6 which also means there is actually a big difference in the behavior of these Eddies. Large eddies will have a different behavior than small ones.

In turbulent flows, eddies can be categorized into large and small scales. Large eddies are associated with the larger structures of flow and can capture more energy from the mean flow, influencing the overall dynamics significantly. Small eddies, meanwhile, behave more uniformly and are typically isotropic. Their characteristics are largely independent of the flow's geometry. The way energy is transferred between these two scales is crucial in understanding turbulence. Large eddies draw their energy from the mean flow, while small eddies extract energy from larger eddies, creating an energy cascade in turbulence.

Imagine a team sports game where large players dominate the field with their strength and strategy (large eddies), while smaller players navigate quickly around them (small eddies). The larger players set the pace and rhythm of the game, while the smaller players adapt and respond to those changes. This mirrors how large eddies influence the flow and how smaller eddies react to those larger movements.

The important thing to remember is that large eddies extract energy from the mean flow, whereas small eddies take energy from a little bit larger eddies which take more energy from the larger eddies than them. So energy is in the form of a cascade. This term is called the Kolmogorov hypothesis.

The Kolmogorov hypothesis suggests that in a turbulent flow, energy passes from larger scales of eddies to smaller ones in a systematic manner, a process known as the energy cascade. Large eddies, which are powerful and have massive energy, interact with mean flow, while small eddies draw energy from these large ones. The interplay between these two sizes of eddies helps form the chaotic structures seen in turbulence, giving insight into the energy distribution and dynamic behavior of turbulent flows.

Consider a waterfall where the energy of the water cascades down from the top (large eddies) to the rocks and small pools below (small eddies). Just as large amounts of water crash down and create smaller splashes and movements in the water below, large eddies transfer their energy in a cascade effect, leading to smaller disturbances.

In LES, the larger eddies are computed with a time-dependent simulation where the influence of small eddies is incorporated through a turbulence model. We solve for large eddies directly and account for small eddy effects through modeling techniques.

In LES, we focus directly on solving for large eddies, which means we can use time-dependent simulations to understand their behavior better. The influence of small eddies, which are too numerous and small to model individually, is included through a turbulence model. This allows simulations to capture the significant behaviors of turbulent flow without the computational burden of resolving every small scale directly.

Think of cooking a complex recipe where the main ingredients (large eddies) are the center of your attention. You monitor their cooking closely, while seasonings and spices (small eddies) are added in with general guidelines—not measured precisely, but still impacting the overall flavor. Just like the recipe comes together from focusing on the main elements, LES builds an accurate picture of turbulence by studying large scales while modeling the small scales.

LES uses spatial filtering operations to separate large and small eddies. The filter width is set to be close to the size of the mesh. The scales directly solved for on the grid are called grid scales for large eddies, and for the smaller ones, they are called subgrid scales (SGS).

In LES, a filtering operation separates large eddies from small eddies to facilitate a more manageable computation. The filter width is chosen based on the size of the computational mesh. This allows the simulation to focus on capturing the dynamics of larger scales while smaller ones are approximated through subgrid scale models. The governing equations are adjusted accordingly to reflect these differences, ensuring that the model results are coherent and useful for predicting flow behavior.

Imagine using a sieve to separate large chunks of fruit from juice—what remains in the sieve (large eddies) is what you want to focus on while some smaller fruit particles pass through (small eddies). Just like the juice flows through to be processed differently than the solid bits, in LES, larger scales are captured while simpler models are used to predict the impact of smaller scales.

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