# Using Lattice BGK Models for Navier-Stokes Equation Derivation

### Key Takeaways

• What are lattice Boltzman methods?

• Lattice Boltzman model advantages and applications.

• Substituting the lattice BGK model for the Navier-Stokes equation. Use of lattice BGK model for turbulent flow analysis. Image source.

Mathematical algorithms are typically developed in response to the lack of readily available tools to solve a particularly challenging problem. In some cases, however, solution paradigms or models are created based on the existence of specific tools. Lattice Boltzman methods ( LBM) are examples of the latter, as LBM was specifically created to leverage the capabilities of massively parallel processing computer environments, such as supercomputers.

Today, virtually all computing platforms have some degree of parallelism built in. This may include multi-core microprocessors and/or graphical processing units (GPUs) that greatly increase mathematical accuracy for complex problem solutions and reduce computational time expense. Both of these attributes promote the use of alternative lattice BGK models for the Navier-Stokes equation when performing fluid dynamics analysis.

## What Are Lattice Boltzman Methods?

Typical CFD methods seek to explain the behavior of fluid properties such as momentum and energy from a macroscopic vantage point, often at the boundary layer between a surface and the fluid environment. Lattice Boltzmann methods, on the other hand, utilize fictitious or fictive particles on a much smaller scale. This allows for discretization over a defined lattice and the application of parallel processing for solutions of flow propagation and internal collisions.

This structure allows for variation of fluid parameters between nodes of the lattice. For example, fluid density may change as fluid propagates from point to point in the lattice indicative of flow collision and streaming activity. This activity defines what is known as the Bhatnagar Gross and Krook (BGK) or lattice Boltzman BGK model, which offers several advantages for CFD analysis.

## Lattice BGK Model Advantages and Applications

The image above illustrates the application of lattice BGK modeling to analyze the creation of vortices. This coupled with vortex shedding is a major area of turbulent fluid flow study where BGK models can be utilized. Lattice Boltzman methods, and in particular BGK models, have several positive attributes (listed below) that make them well-suited for a large number of fluid flow problems where molecular level behavior is of interest; more traditional CFD techniques have difficulty solving these problems, at least to a high degree of accuracy.

### The Advantages of Lattice BGK Models

• ★ Ease of incorporation into a parallel computing architecture
The use of a discrete model and a lattice architecture promotes the development of parallel algorithms that can significantly reduce processing time.
• ★ Complex geometry analysis at the molecular level
The ability to view fluid particles at the molecular level allows for more complex geometries and greater accuracy of fluid parameters at the macroscopic level.
• ★ Enables more advanced study of fluid flows
The ability to break liquid and gas flows into droplets opens up a new level of understanding of the internal behavior within fluids.

These attributes make using the alternative lattice BGK models for Navier-Stokes equations an option worth exploring for fluid flow analysis.

## Substituting the Lattice BGK Model for the Navier-Stokes Equation

Fluid flow analysis for aeronautical analysis often involves the creation of high-order mesh grids using algorithms such as Delauney triangulation. BGK models employ a simple lattice structure that can be constructed using a small portion of the processing time required for complete solutions, which accentuates one of the best reasons to use lattice BGK models. This coupled with the ability to study compressible and incompressible fluid flow parameters in greater detail make using this alternative to the traditional Navier-Stokes equations an attractive solution method.

Whether you choose to substitute lattice BGK models for the Navier-Stokes equation in your CFD analysis or not, you should utilize advanced CFD solver tools as provided by Cadence for your fluid flow analysis.