Week 9 - FVM Literature Review
Finite Volume Method (FVM):-
The Finite Volume Method (FVM) is a numerical technique used for solving partial differential equations (PDEs) in computational fluid dynamics (CFD) and other fields of computational science and engineering. It is particularly well-suited for problems involving conservation laws, such as fluid flow and heat transfer. FVM discretizes the domain into small control volumes (or cells) and focuses on the conservation of quantities within these volumes. It can be summarized as follows:
1. Discretization:- The continuous PDEs governing a physical system are discretized by dividing the domain into a finite number of control volumes. These control volumes can be of various shapes, including cubes, hexahedra, triangles, or tetrahedra in 3D, and rectangles or triangles in 2D.
2. Integral Form:- The integral form of the governing equations is employed. This means that the equations are integrated over each control volume to obtain discrete equations that relate fluxes at the faces of the control volumes to changes in the conserved quantities within the volumes.
3. Conservation Principle:- FVM is based on the principle of conservation. It ensures that the net flow of a conserved quantity into or out of a control volume equals the rate of change of that quantity within the volume.
4. Interpolation and Flux Calculation:- To calculate fluxes at the faces of control volumes, interpolation schemes are often used to estimate values at cell faces. These fluxes represent the transport of conserved quantities across the control volume boundaries.
5. Solution Procedure:- The resulting system of algebraic equations is solved iteratively using numerical methods, such as the SIMPLE or PISO algorithm for fluid flow problems.
Differences between Finite Difference Method (FDM) and Finite Volume Method (FVM):-
1. Control Volume vs. Grid Points:- In FDM, the domain is discretized using a grid, where values are assigned to grid points (nodes). In contrast, FVM uses control volumes, where values represent averages over the volume. FVM focuses on the conservation of quantities within these volumes.
2. Conservation Principle:- FVM inherently enforces the conservation principle, making it well-suited for problems involving conservation laws. FDM does not have this property by default.
3. Treatment of Boundaries:- FDM often requires special treatments at boundaries to approximate derivatives. FVM handles boundary conditions naturally by considering the flow of quantities across control volume faces.
4. Flexibility:- FVM is more versatile in handling complex geometries because it can use non-structured grids with control volumes of different shapes. FDM is typically more suited for regular grids.
Need for Interpolation Schemes and Flux Limiters in FVM:-
In FVM, interpolation schemes and flux limiters are essential to estimate values at control volume faces and manage numerical instabilities.
1. Interpolation Schemes:- To calculate the heat flux at the faces of control volumes, FVM requires estimating the temperature gradient. Common interpolation schemes, like linear or quadratic interpolation, are used to estimate temperature values at the control volume faces based on the values at cell centers. This interpolation ensures accurate flux calculations.
2. Flux Limiters:- In cases where steep gradients occur, such as near discontinuities or shock waves, standard FVM may lead to numerical oscillations. Flux limiters are employed to restrict the magnitude of the flux to prevent unphysical behavior. They ensure that the calculated flux respects the physical bounds of the problem, enhancing stability and accuracy.
In short, interpolation schemes help in estimating values at control volume faces, and flux limiters are used to control the flux magnitude to maintain stability and accuracy in FVM simulations, particularly when dealing with complex or discontinuous phenomena.