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.

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

Direct Numerical Simulation (DNS) solves the time dependent Navier-Stokes equations, resolving from the largest length
scale of a computational domain size to the smallest length scale of turbulence eddy (Kolmogorov length scale).

Three-dimensional industrial scale problems are concerned with the time averaged (mean) flow, not the instantaneous motion.
The preferred approach is to model turbulence using simplifying approximations, and not resolve it.

All of the previously described models are incapable of predicting boundary layer transition. To include the effects of
transition additional equations are necessary.

While Reynolds-averaged Navier-Stokes (RANS) resolves the mean flow and requires a turbulence model to account for the
effect of turbulence on the mean flow, Large Eddy Simulation (LES) computes both the mean flow and the large energy containing
eddies.

The Smagorinsky-Lilly SGS model is based on the Prandtl’s mixing length model and assumes that a kinematic SGS viscosity
can be expressed in terms of the length scale and the strain rate magnitude of the resolved flow.

Recognizing variations in space and time, Germano et al. (1991) proposed the dynamic model to compute the value of rather than specifying it explicitly.

For internal wall bounded flows, proper mesh resolution is required in order to calculate the steep gradients of the velocity
components, turbulent kinetic energy, dissipation, as well as the temperature.

During the past decades turbulence models of various complexities have been developed. Turbulence models that employ the
most assumptions are typically the least demanding from a CPU cost standpoint.

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

Recognizing variations in space and time, Germano et al. (1991) proposed the dynamic model to compute the value of rather than specifying it explicitly.

Recognizing ${C}_{s}$ variations in space and time, Germano et al. (1991) proposed the
dynamic model to compute the value of ${C}_{s}$ rather than specifying it explicitly.

It is implemented by utilizing two filters: a cutoff filter $\text{\Delta}$ and a test (coarse) cutoff filter $\stackrel{\sim}{\Delta}$.

The subgrid stress tensor ${\tau}_{ij}^{\text{'}}$ with the cutoff filter ($\text{\Delta}$) is: ${\tau}_{ij}^{\text{'}}=\rho \tilde{{u}_{i}{u}_{j}}-\rho \tilde{{u}_{i}}\tilde{{u}_{j}}=-2\rho {\left({C}_{S}\text{\Delta}\right)}^{2}\left|\tilde{S}\right|\tilde{{S}_{ij}}$. Where $\left|\tilde{S}\right|=\sqrt{2\tilde{{S}_{ij}}\tilde{{S}_{ij}}}$ is the strain rate magnitude.

Figure 1 shows a resolved turbulence region
utilizing Large Eddy Simulation (LES) and a modeled region assuming the subgrid tensor ${\tau}_{ij}^{\text{'}}$.

The test subgrid stress tensor ${T}_{ij}$ with the coarse filter ($\stackrel{\sim}{\Delta}$) can be written as (1)

$\left|\tilde{\tilde{S}}\right|=\sqrt{2\tilde{\tilde{{S}_{ij}}}\tilde{\tilde{{S}_{ij}}}}$ is the coarse filtered strain rate magnitude.

$\widehat{\tilde{{S}_{ij}}}=\frac{1}{2}\left(\frac{\partial \widehat{\tilde{{u}_{i}}}}{\partial {x}_{j}}+\frac{\partial \widehat{\tilde{{u}_{j}}}}{\partial {x}_{i}}\right)$ is the filtered strain rate tensor, using the coarse
cutoff filter.

Figure 2 shows a resolved
LES region and a corresponding subgrid modelled region (T) when the coarse filter is
employed.

Because of the coarse filtering, the test (coarse) subgrid stress tensor ${T}_{ij}$ should be a summation of the coarse filtered subgrid stress
tensor $\tilde{{\tau}_{ij}^{\text{'}}}$ and the Leonard stress tensor ${L}_{ij}$. (2)

${L}_{ij}$ is the Leonard subgrid stress tensor, representing the
contribution to the subgrid stresses by turbulence length scales smaller than the test
filter but larger than the cutoff filter.

The Leonard subgrid stress tensor can be arranged as (4)

where ${M}_{ij}=2\rho {\text{\Delta}}^{2}\left(\tilde{\left|\tilde{S}\right|\tilde{{S}_{ij}}}-{\alpha}^{2}\left|\tilde{\tilde{S}}\right|\tilde{\tilde{{S}_{ij}}}\right)$.

Since the above equation is overdetermined a minimum
least square error method is used to determine the coefficient ${C}_{s}$. (6)

In order to avoid numerical instabilities associated with the above
equation, as the numerator could become negative, averaging of the error in the minimization
is employed. (7)