Introduction of background knowledge regarding flow physics and CFD as well as detailed information about the use of AcuSolve and what specific options do.

This section on basics of fluid mechanics covers topics describing the fundamental concepts of fluid mechanics, such as
the concept of continuum, the governing equations of a fluid flow, definition of similitude and importance of non-dimensional
numbers, different types of flow models and boundary layer theory.

This section on turbulence covers the topics describing the physics of turbulence and turbulent flow. It also covers the
modeling of turbulence with brief descriptions of commonly used turbulence models.

This section on numerical approximation techniques covers topics, which describe the numerical modeling of the fluid flow
equations on a computational domain, such as spatial discretization using finite difference, finite element and finite volume
techniques, temporal discretization and solution methods.

The governing equations, that is, the Navier – Stokes equations in continuum mechanics are a set of coupled non–linear
partial differential equations derived from the conservation laws for mass, momentum and energy.

This section describes the formulation and methodology of finite volume method to solve the governing equations on a computational
domain. It also describes the cell centered and face centered approaches used for finite volume formulation.

The finite element approach is based on the discretization of the domain into a set of finite elements, which are usually
triangles or quadrilaterals in a two dimensions and tetrahedral, hexahedra, pyramids or wedges in three dimensions, and
using variational principles to solve the problem by minimizing an associated error function or residual.

This section describes the various approaches used to discretize the governing equation in the temporal domain such as
two step, multistep and multistage methods.

This section describes the direct and iterative solution methods used to solve the linear system of equations obtained
after spatial and temporal discretization of the governing equations.

This section on AcuSolve solver features covers the description of various solver features available in AcuSolve such as heat transfer, fluid structure interaction and turbulence modeling.

Collection of AcuSolve simulation cases for which results are compared against analytical or experimental results to demonstrate the accuracy
of AcuSolve results.

Introduction of background knowledge regarding flow physics and CFD as well as detailed information about the use of AcuSolve and what specific options do.

This section on numerical approximation techniques covers topics, which describe the numerical modeling of the fluid flow
equations on a computational domain, such as spatial discretization using finite difference, finite element and finite volume
techniques, temporal discretization and solution methods.

This section describes the formulation and methodology of finite difference method to
solve the governing equations on a computational domain.

The finite difference method is the oldest method for the numerical solution of partial
differential equations. It is also the easiest to formulate and program for problems which
have a simple geometry.

When calculating derivatives finite difference replaces the infinitesimal limiting process
with a finite quantity. The derivative of a function at point expressed as: (1)

The preceding expression is commonly referred to as the forward difference approximation.
This derivative can have more refined approximations using a number of approaches such as
truncated Taylor series expansions and polynomial fitting.

The term $O\left(\Delta x\right)$ gives an indication of the magnitude of the error as a
function of the mesh spacing and is therefore termed as the order of magnitude of the finite
difference method. In the above formulation, the finite difference approximation employed is
first-order accurate. A second-order approximation would have the order of magnitude
expressed as $O\left(\Delta {x}^{2}\right)$. Most of the finite difference methods used in practice are
second-order accurate. An example of a second-order method is the central difference
approximation of the first derivative. It is expressed as: (3)

The basic methodology in the simulation process using finite difference approach consists
of the following steps.

The finite difference formulation generally employs a structured grid. The most commonly
used indices are $i$, $j$ and $k$ for the grid lines at $x={x}_{i}$, $y={y}_{j}$ and $z={z}_{k}$, respectively. The function value at such a grid point is
expressed as ${f}_{i,j,k}\equiv {f}_{ijk}\equiv f\left({x}_{i},{y}_{j},{z}_{k}\right)$.

Consider the one dimension advection-diffusion equation for a scalar quantity $\phi $ governed by (4)

where $u$ is the specified velocity, $\rho $ is the density of the fluid and $\u03f5$ is the diffusivity.

If the domain is defined by the boundary $x=0$ and $x=L$ and the boundary conditions by $\phi \left(0\right)={\phi}_{0}$ and $\phi \left(L\right)={\phi}_{L}$, the domain can be discretised (non-uniformly) by a total of $N+1$ grid points for the finite difference solution of the
problem.

The diffusion term can be approximated using the central difference (both the inner and
outer derivative) as: (5)

If the grid is uniform and the values of density, velocity and coefficient of diffusion are
constant throughout the domain the above equations reduce to: (8)

The resulting set of linear algebraic equations can be solved to get the values of at the
grid points.

Finite difference methods are advantageous for the numerical solution of partial
differential equations because of their simplicity, efficiency and low computational cost.
Their major drawback is their geometrical inflexibility, such as application on an
unstructured grid or moving boundaries. Their formulation increases in complexity as the
complexity of the domain increases.

The restrictions resulting from the above mentioned geometrical complexities can be
alleviated by the use of methods such as grid transformation and immersed boundary
techniques.