Week 9 - FVM Literature Review
FVM Literature Review
- The Finite Volume Method (FVM) is a widely used numerical technique for solving partial differential equations (PDEs), particularly in the field of computational fluid dynamics (CFD) and heat transfer problems.
- It provides an effective approach for simulating physical processes in engineering and scientific applications.
Finite Volume Method:
The Finite Volume Method discretizes the domain of interest into a finite number of control volumes, each of which stores a discrete value representing the average behavior of the underlying PDE within that volume.
The method is conservative, ensuring that integral properties of the governing equations are preserved within each control volume.
Significance of FVM:
- Discretization:
The domain is divided into control volumes, and the governing equations are converted into integral form over each volume, resulting in a set of algebraic equations.
Spatial discretization is then performed using numerical integration techniques.
- Conservation Principles:
FVM employs key conservation principles, such as mass, momentum, and energy, and thus well-suited for modeling fluid flow and heat transfer problems.
The method accurately captures flow patterns and temperature distributions within the discretized domain.
- Applicability to Complex Geometries:
FVM is highly adaptable to complex geometries since it does not require a structured grid.
Irregularly shaped control volumes can be used, allowing simulations in realistic geometries without compromising numerical accuracy.
- Flexibility in Boundary Conditions:
FVM handles various boundary conditions efficiently, including Dirichlet, Neumann, and mixed conditions.
By applying different interpolation and extrapolation schemes, accurate results can be obtained at boundaries.
Differences between FVM and FDM
- WRT Spatial Discretization:
FVM: The primary unknowns are located at the control volumes. The spatial derivatives are computed using fluxes across the cell faces, leading to the conservative nature of the method.
FDM: The primary unknowns are usually located at discrete grid points. The spatial derivatives are approximated using finite differences at these points. FDM does not inherently guarantee conservation properties.
- WRT Numerical Dissipation and Dispersion:
FVM: FVM exhibits lesser numerical dissipation and dispersion compared to FDM. It captures sharp gradients and discontinuities in the solution in better way, making it suitable for problems with shocks or steep fronts.
FDM: FDM tends to introduce more numerical dissipation and dispersion, which can smoothen sharp features and reduce accuracy in resolving discontinuities.
- WRT Boundary Treatment:
FVM: FVM naturally handles complex boundary geometries by employing appropriate interpolation schemes and flux limiters at the cell faces.
FDM: FDM might require special treatment near boundaries to maintain accuracy, especially for non-structured grids or complex boundary shapes.
- WRT Treatment of Irregular Domains:
FVM: FVM efficiently deals with unstructured grids, enabling simulations in complex geometries without the need for structured meshing.
FDM: FDM typically relies on structured grids, which might limit its suitability for irregular domains.
The 1D linear heat conduction equation:
∂T/∂t = α * ∂x2/∂2 T
where
T is the temperature distribution,
t is time,
x is the spatial coordinate, and
α is the thermal diffusivity.
Interpolation schemes
Two common interpolation schemes are linear interpolation and limited interpolation.
- Linear Interpolation:
Linear interpolation provides a simple and straightforward estimation of the values at cell faces. However, it can lead to wild oscillations when sharp gradients or discontinuities are present in the solution. It lacks the necessary mechanism to prevent overshooting or undershooting of the true solution.
- Limited Interpolation:
Limited interpolation schemes are designed to address the issue of wild oscillations caused by linear interpolation. These schemes use slope limiters to restrict the gradients near sharp features.
Popular limiters like van Leer, MinMod, and Superbee are used to limit the slopes effectively and maintain the monotonicity of the solution. This prevents unphysical oscillations and ensures stability and accuracy.
Flux limiters
Flux limiters are crucial components in FVM that govern the computation of fluxes at the cell faces. The flux limiters control the contribution of neighboring cells to the overall flux, especially when sharp gradients or discontinuities occur.
- Role of Flux Limiters:
In regions with smooth variations, flux limiters behave like first-order upwind schemes, providing stability and robustness. However, in regions with steep gradients or shocks, the flux limiters activate, modifying the flux computation to capture the sharp features more accurately.
- Flux Limiter Types:
Various flux limiters are available, each with specific properties suitable for different scenarios. Some popular flux limiters include the Van Leer limiter, MinMod limiter, Superbee limiter, and more.
Need for interpolation schemes
The need arises as the values of the solution are only known at the centers of the finite volumes.
However, the values of the solution on the faces of the finite volumes are needed to compute the fluxes.
This is where interpolation schemes come in. Interpolation schemes are used to estimate the values of the solution on the faces of the finite volumes from the known values at the centers of the finite volumes.
The choice of interpolation scheme depends on the smoothness of the solution.
If the solution is smooth, then a simple interpolation scheme, such as linear interpolation, may be sufficient.
However, if the solution is not smooth, then a more complex interpolation scheme, such as quadratic interpolation or cubic interpolation, may be needed.
Need for flux limiters
The need arises when the numerical oscillations can occur near discontinuities in the solution. These oscillations can be caused by the use of high-order interpolation schemes or by the use of a coarse grid.
- Flux limiters are used to prevent these oscillations by adjusting the values of the fluxes at the faces of the finite volumes.
- Flux limiters work by limiting the amount of numerical diffusion that can occur at the faces of the finite volumes.
This is done by adding a correction term to the fluxes that is proportional to the gradient of the solution. The correction term is designed to reduce the amount of numerical diffusion, but it does not affect the solution in smooth regions.
Conclusion:
Interpolation schemes and flux limiters are essential tools for the FVM. They are used to estimate the values of the solution on the faces of the finite volumes and to prevent numerical oscillations, respectively. These tools are essential for ensuring that the FVM produces accurate and stable solutions to PDEs.