2.1. Large and Small Eddies

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.

In computational fluid dynamics, Large Eddy Simulation (LES) is a technique that balances accuracy and computational efficiency. Direct Numerical Simulation (DNS) provides the best accuracy but requires extensive computational resources. In contrast, Reynolds Averaging simplifies the equations but sacrifices accuracy. LES aims to find a middle ground by modeling larger eddies directly while approximating the smaller eddies.

Think of LES as a chef who prepares a gourmet meal. DNS would be the chef meticulously measuring every single ingredient to perfection, which takes a lot of time. Reynolds averaging is like throwing everything into the pot without measuring carefully, which may taste good but lacks precision. LES is the chef who measures the key ingredients accurately while using some shortcuts for the less critical components.

So there is a big difference in the behaviors of large and small eddies in turbulent flow fields. Large eddies are more anisotropic, and their behavior is dictated by the geometry of the problem domain and the boundary conditions. The larger eddies must also depend upon the body forces acting, whereas small eddies are nearly isotropic and have a universal behavior as demonstrated by Kolmogorov.

In turbulent flows, large eddies behave differently from small eddies. Large eddies are influenced by the shape of their surrounding environment (geometry) and the conditions at boundaries (boundary conditions). They also respond to external forces, such as gravity. In contrast, small eddies are isotropic, meaning their properties are uniform in all directions, and they follow universal behaviors as per Kolmogorov's theories.

Imagine a large storm cloud (large eddy) shaped by nearby mountains (geometry) affecting its shape and movement. The storm cloud’s behavior is also influenced by wind (body forces). In comparison, think of a small whirlpool in a sink (small eddy) - its rotation and behavior are similar regardless of the sink’s shape. It behaves uniformly and is less influenced by external factors.

Important thing to remember is that the large eddies extract energy from the mean flow, so, larger, that is more energy will be there, and they take energy from the mean flow that the flow real flow. Whereas small eddies take energy from a little bit larger eddies which takes more energy from the larger eddies than them.

Large eddies play a crucial role in energy dynamics within turbulent flows. They derive energy from the overall flow (mean flow) and transfer some of this energy down to smaller eddies, which in turn extract energy from these larger eddies. This process is known as energy cascading, where energy moves from larger to smaller scales, contributing to the overall turbulence.

Consider a waterfall (representing the mean flow) that powers a large waterwheel (large eddy). The energy from the waterfall turns the wheel and powers smaller mechanisms, like a millstone (small eddy), that grind grain. The large flow (waterfall) provides energy to the wheel, which then passes some of that energy to the smaller mechanisms.

This is actually a major hurdle in the path of research just trying to develop a universal turbulence model, which is what is the hurdle: the variation of eddies from large scale to small scale? And the fact that the largest scale eddies are not universal in nature; only smaller eddies are a single turbulence model must be able to describe the collective behavior of all the eddies.

Developing a single turbulence model that can accurately describe all scales of eddies is a significant challenge in fluid dynamics. The behavior of large eddies varies greatly based on their environment, while smaller eddies follow more uniform patterns. This variance complicates the formulation of universal models that can accommodate all types of turbulence behavior consistently.

Imagine trying to create a set of rules for different sports. Football (large eddies) may have varying strategies depending on the field size, while chess (small eddies) has more standardized moves. Creating one rulebook to effectively cover both types of games presents a challenge because of their differing complexities and rules.

In LES, the larger eddies are computed with a time-dependent simulation where the influence of the small eddies is incorporated through a turbulence model. So, we say in LES two parts large eddies and there are small eddies.

Large Eddy Simulation (LES) calculates the motion of larger eddies while estimating the influence of smaller eddies through a model. This hybrid approach allows for efficient computation of turbulent flows by explicitly handling large-scale structures and using models for smaller-scale fluctuations.

Think of a film production where the director (LES) decides to showcase the main actors (large eddies) prominently while using background actors (small eddies) that follow a script (turbulence model) to enhance the storyline without being in the spotlight. This keeps the focus on the main plot while still acknowledging the role of supporting characters.

LES uses spatial filtering operation to separate the large and the small eddies as I have told you, and the filtered Navier-Stokes equations are used as the governing equation for large Eddy simulation.

To distinguish between large and small eddies in LES, a spatial filtering operation is employed. This filtering reveals the significant motions of larger eddies while limiting the influence of smaller scales. The equations governing the large eddies are then modified accordingly to account for this separation.

Imagine using a colander (filter) to drain pasta. The colander allows the large pasta pieces to pass through while holding back smaller particles (small eddies). In a similar way, the filtering operation captures the important large-scale behaviors in fluid motion while minimizing the impact of less significant motions.

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 ones, the Subgrid Scales (SGS). So, small scales are not captured by those grid correct because they are larger in size to capture those eddies.

In LES, we distinguish between grid scales (GS) and subgrid scales (SGS). Grid scales refer to the scales of motion that can be directly resolved by the computational mesh used in the simulation, primarily for large eddies. Subgrid scales encompass the smaller eddies that are not resolved directly but are modeled instead.

Think of a painter working on a large mural (grid scales) where only the major subjects are fully painted. The small details (subgrid scales) might be suggested with broader strokes rather than meticulously painted. This method allows for a focus on the overall picture while providing an impression of finer details.

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