2.5.2.1.3. Finite Volume Method

The most common discretization techniques available for the numerical solution of partial differential equations are the finite difference method, the finite element method, and the finite volume method.

In computational fluid dynamics (CFD), the flow fields are represented in discrete forms so that complex equations can be solved numerically. The three primary techniques for this discretization are:

1. Finite Difference Method (FDM): This technique approximates derivatives by evaluating values at discrete grid points.
2. Finite Element Method (FEM): This involves dividing the flow domain into smaller elements (or volumes) and solving equations over these small sections.
3. Finite Volume Method (FVM): Like FEM, this method divides the domain but focuses on the flow across volumes and conservation across these volumes, making it particularly useful for fluid flow problems.

Imagine you're trying to measure the depth of a lake. Instead of diving deep and trying to get an exact reading, you might throw a series of ropes or sticks into the water at various points along the shore. Each point represents a small section or volume of the lake, giving you a good approximation of its depth throughout without needing to measure every single spot.

In finite element or finite volume method the flow field is broken into smaller fluid elements called cells. Alright. So, for 2D domains cells are areas, and for 3D domains, these cells are volumes. So, the differential equations are written in appropriate form for each element and this set of resulting algebraically equations are solved numerically.

Under the Finite Volume Method (FVM), the computational domain is divided into distinct volumes or cells. This division allows for the application of the conservation laws (mass, momentum, energy) across these volumes. When the flow across the boundaries of each cell is considered, equations governing the fluid behavior can be integrated over each cell. The resulting equations are often a set of algebraic equations that are more manageable and can be solved using numerical methods.

Consider a water distribution system in a city. Each neighborhood can be thought of as a cell. The flow of water into and out of each neighborhood (cell) must be balanced to ensure everyone receives their fair share. By analyzing the flow at the entrance and exits of each neighborhood, we can better understand and manage the entire city's water distribution system.

This is an example where you see it is a 2D domain and therefore this is a cell. In the computational domain, it is 2D. This is 3-dimensional computational domain. And you see, these are like volumes.

In the context of the finite volume method, the flow domain can be represented in two or three dimensions. In 2D, each 'cell' represents an area over which calculations are made, while in 3D, each 'cell' represents a volume. These cells serve as the framework for applying the conservation equations. The fluid flow characteristics such as velocity and pressure are calculated at different points within these cells, and interactions between cells help in understanding the overall flow behavior.

Think of a garden divided into plots (cells): each plot can represent different plants that interact with water and nutrients. As you analyze each plot individually while observing how water flows between them, you understand how best to manage the overall garden's irrigation.

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