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 physics of turbulence introduces a brief history of turbulence and covers the theory behind turbulence
generation, turbulence transition and energy cascade in fluid flows.

The Reynolds number is not only used to characterize the flow patterns, such as laminar or turbulent flow, but also to
determine the dynamic similitude between two different flow cases.

The physics of turbulent flows have been discussed by presenting experimental observations and comparing it to laminar
flows. In this chapter, the focus will shift to the governing equations of these flow fields.

Turbulence is composed of turbulent eddies of different sizes. At high Reynolds numbers, a scale separation exists between
the largest eddies and smallest eddies.

This section covers the numerical modeling of turbulence by various turbulence models, near wall modeling and inlet turbulence
parameters specified for 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.

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 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 physics of turbulence introduces a brief history of turbulence and covers the theory behind turbulence
generation, turbulence transition and energy cascade in fluid flows.

The U term becomes $\nabla \times \frac{\partial \overrightarrow{u}}{\partial t}=\frac{\partial}{\partial t}\left(\nabla \times \overrightarrow{u}\right)=\frac{\partial \overrightarrow{\omega}}{\partial t}$

The P term vanishes as $\nabla \times \nabla \cdot \text{p}=0$

The V term becomes $\nabla \times \left(\frac{\mu {\nabla}^{2}\overrightarrow{u}}{\rho}\right)=\frac{\mu}{\rho}{\nabla}^{2}\overrightarrow{\omega}$

The C term becomes $\nabla \times \left(\overrightarrow{u}\cdot \nabla \overrightarrow{u}\right)=\nabla \times \frac{1}{2}\nabla \left({u}^{2}\right)-\nabla \times \left(\overrightarrow{u}\times \overrightarrow{\omega}\right)=\nabla \times \left(\overrightarrow{\omega}\times \overrightarrow{u}\right)$

The C term can be rearranged as $\nabla \times \left(\overrightarrow{\omega}\times \overrightarrow{u}\right)=\left(\overrightarrow{u}\cdot \nabla \right)\overrightarrow{\omega}-\left(\overrightarrow{\omega}\cdot \nabla \right)\overrightarrow{u}$

After substituting above terms, the vorticity transport equation can be obtained as:
(5)

The vortex stretching term, ${\omega}_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}$, on the right-hand side of the equation corresponds to the
enhancement of vorticity ${\omega}_{j}$ when a fluid element is stretched $\frac{\partial {u}_{i}}{\partial {x}_{j}}$<0. In other words, when the cross section of the fluid
element is decreased, the vorticity is increased. The image below shows the concept of
vortex stretching. In the image, there are two cylindrical fluid elements within the
streamwise fluid flow. The concept shows that compared to a thick element a thin element has
stronger vorticity due to the angular momentum conservation.

In two-dimensional flow, the vortex stretching term ${\omega}_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}$ vanishes since ${\omega}_{x}=0$, ${\omega}_{y}=0$ and $\frac{\partial}{\partial z}=0$. Therefore, the vortex stretching term is essential to the
energy cascade for three-dimensional turbulent flow.