This document compares three popular discretization approaches in computational fluid dynamics (CFD): finite difference method (FDM), finite volume method (FVM), and finite element method (FEM). It explains that FDM replaces derivatives with difference formulas on a grid, FVM is based on integral forms of partial differential equations solved over finite volumes, and FEM uses piecewise basis functions over finite elements. While each method has advantages and disadvantages, FVM generally provides better conservation properties and can handle complex domains, though all three approaches make the modeling of fluid flows and related phenomena possible through computer-based simulation.

1. Approaches in CFD
Author : Parham Sagharichi Ha

2. Agenda
▪ Computational Fluid Dynamics or CFD
▪ Applications, Advantages, disadvantages of CFD
▪ Process in CFD
▪ Popular discretization approaches in CFD
▪ Finite difference method (FDM)
▪ Finite volume method (FVM)
▪ Finite element method (FEM)

3. Computational Fluid Dynamics
▪ Computational Fluid Dynamics or CFD is the analysis of systems
involving fluid flow, heat transfer and associated phenomenas
such as chemical reactions by means of computer based
simulation.

4. Applications of CFD
▪ Aerodynamics of aircraft and vehicles: lift and drag.
▪ Hydrodynamics of ship.
▪ Power Plant: Gas turbines.
▪ Chemical Process Engineering: mixing and separation.
▪ Marine engineering: loads on off-shore structure.
▪ Environmental Engineering: Distribution of pollutants
▪ Biomedical Engineering: blood flows through arteries
and veins.

5. Advantages of CFD
▪ Study specific terms in the governing equations in a detailed
fashion.
▪ Complements experimental and analytical.
▪ CFD substantially reduces lead times and cost in design.
▪ Simulating flow conditions that are not reproducible in
experimental test.
▪ Provide rather detailed, visualized and comprehensive information
as compared to other methods.

6. Disadvantages of CFD
▪ A large amount of processing power is nedded to run some test
case.
▪ If the processing of reading & writing to the CFD packages
slow,then the whoule solution process is slowed down.

7. Process in CFD
▪ In order to provide easy access to, almost all current commercial
and some shareware CFD package include user friendly Graphical
User Interface(GUI) applications. The codes that provide a
complete CFD analysis consists of three main elements :
1. Pre-Processor.
2. Solver.
3. Post-Processor.

8. Pre - Processor
▪ Pre-Processing consists of input to flow problem by user and
transformation to a form usable by the solver.
1. Defining geometry of interest; Computational Domain.
2. Sub Division of domain into a number of overlapping sub
domain; a grids(mesh) cell.
3. Selection of physical and chemical phenomena that needs to
be modeled.
4. Defining fluid properties.
5. Specifying of Boundary Conditions.

9. Solver
▪ In outline the numerical methods that form the
basis of the solver perform the following steps:
1. Approximation of unknown flow variables by
means of functions.
2. Discretization and mathematical manipulations.
3. Solutions to algebraic equations.

10. Post - Processor
▪ Modern CFD packages have outstanding data visualization tools, which
includes:
1. Domain geometry and grid display.
2. Vector plots.
3. Lines and shaded contour plots.
4. 2D and 3D surface plots.
5. Particle Tracking.
6. View manipulation.
7. Colour Post script output.

11. Popular discretization approaches in CFD
▪ Finite Difference Method
▪ Finite Volume Method(ANSYS FLUENT)
▪ Finite Element Method

12. Finite difference method (FDM)
▪ FDM is created from basic definition of differentiation that is
𝑑𝑓
𝑑𝑥
=
𝑓 𝑥 + ℎ − 𝑓(𝑥)
ℎ
▪ In numerical analysis, its not possible to divide a number by "0" so
"zero" means a small number. So FDM is similar to differential
calculus but it has killed the heart that is limit tenda to "zero". So
in most of the cases accuracy of FDM increases with refining grid.
Easy method but not reliable for conservative differential equations
and solutions having shocks. Tough to implement in complex
geometry where it needs complex mapping and mapping makes
governing equation even tougher. Extending to higher order
accuracy is very simple.

13. Finite difference method (FDM)
▪ A finite difference method (FDM) discretization is based upon the
differential form of the PDE to be solved. Each derivative is
replaced with an approximate difference formula (that can generally
be derived from a Taylor series expansion). The computational
domain is usually divided into hexahedral cells (the grid), and the
solution will be obtained at each nodal point. The FDM is easiest to
understand when the physical grid is Cartesian, but through the use
of curvilinear transforms the method can be extended to domains
that are not easily represented by brick-shaped elements. The
discretization results in a system of equation of the variable at
nodal points, and once a solution is found, then we have a discrete
representation of the solution.

14. Finite volume method (FVM)
▪ It is a numerical tool that is borrowed from calculus of variation. There are lot of
types of FEM like point collocation method, sub-domain method etc. Here they
assume some trial function and multiply that trial function with weighting
function . In Galerkins method the trial function itself weighting function.
Different methods follow different ways in weighting. Then this weighting
function is multiplied with trial function then integrated over the control volume
( weak form) and equated to zero (This procedure will differ for different types of
FEM but theme is same). Then we get one set of algebraic equations. Solving that
will give solution. Here we are working only in error and differential equation
some times conservative law may be violated. This method is more accurate than
FVM and FDM. Ideal for linear PDEs, expensive and complex for non-linear
PDEs. Here higher order accuracy is achieved by using higher order basis (i.e)
shape functions. Extending to higher order accuracy is relatively complex than
FVM and FDM. Higher order accurate calculations are expensive in computation
and Mathematical formulation especially for non-linear PDEs. Mostly suitable for
Heat transfer, Structural mechanics, vibrational analysis etc.

15. Finite volume method (FVM)
▪ A finite volume method (FVM) discretization is based upon an integral
form of the PDE to be solved (e.g. conservation of mass, momentum, or
energy). The PDE is written in a form which can be solved for a given
finite volume (or cell). The computational domain is discretized into finite
volumes and then for every volume the governing equations are solved.
The resulting system of equations usually involves fluxes of the conserved
variable, and thus the calculation of fluxes is very important in FVM. The
basic advantage of this method over FDM is it does not require the use of
structured grids, and the effort to convert the given mesh in to structured
numerical grid internally is completely avoided. As with FDM, the
resulting approximate solution is a discrete, but the variables are typically
placed at cell centers rather than at nodal points. This is not always true,
as there are also face-centered finite volume methods. In any case, the
values of field variables at non-storage locations (e.g. vertices) are
obtained using interpolation.

16. Finite element method (FEM)
▪ This is similar to FDM but. It didn't kill the theme of
differentiation because we are integrating the differential equation
over a control volume and discretizing the domain. Since we have
integrated the differential equation discetization is mathematically
a valid one. It can be loosely viewed as FEM but weight here used is
1. Here fluxes are integrated and resultant is set to zero, so flux is
conserved. Can handle almost any PDEs and complex domain.
Interpolation is done from face to centre will reduce the accuracy of
this process. Here accuracy is based on order of polynomial used.
FVM can also produce any order accurate numerical solution similar
to FDM but more expensive than FDM Aero acoustic problems use
FVM about 11th order schemes such schemes are rarely used even in
DNS and LES. Ideal for Fluid mechanics.

17. Finite element method (FEM)
▪ A finite element method (FEM) discretization is based upon a
piecewise representation of the solution in terms of specified basis
functions. The computational domain is divided up into smaller
domains (finite elements) and the solution in each element is
constructed from the basis functions. The actual equations that are
solved are typically obtained by restating the conservation equation
in weak form: the field variables are written in terms of the basis
functions, the equation is multiplied by appropriate test functions,
and then integrated over an element. Since the FEM solution is in
terms of specific basis functions, a great deal more is known about
the solution than for either FDM or FVM. This can be a double-
edged sword, as the choice of basis functions is very important and
boundary conditions may be more difficult to formulate. Again, a
system of equations is obtained (usually for nodal values) that must
be solved to obtain a solution.

18. Result
▪ Comparison of the three methods is difficult, primarily due to the
many variations of all three methods. FVM and FDM provide
discrete solutions, while FEM provides a continuous (up to a point)
solution. FVM and FDM are generally considered easier to program
than FEM, but opinions vary on this point. FVM are generally
expected to provide better conservation properties, but opinions
vary on this point also. If you are trying to decide which method to
use, then the best path is probably found by consulting the
literature in the specific problem area.

19. Do you have any questions ?
Parham Sagharichi Ha