Direct numerical simulation has to solve all the turbulence scales from the large eddies down to the smallest Kolmogorov scales. They are based on a three-dimensional and unsteady solution of the Navier-Stokes equations. The size of the calculation domain must be sufficiently large compared to the and the step size of the mesh must be a fraction of the Kolmogorov scale. These requirements will limit the turbulence Reynolds number that can be reached in the simulation.

DNS On the physical intuitive point of view, the Fourier transform operator corresponds to a representation of the fluctuating field as a random superposition of sine and cosine wavy motions characterized by a wave vector and a corresponding spectral density. The direction of represents the direction of evolution of the turbulent eddies whereas the modulus κ or wavenumber is related to the size of eddies (inverse of the eddies length scale).

These complete simulations in which all the eddying scales are resolved, are still limited by computer power, and thus they cannot be applied to practical industrial complex flows. From this point of view an intermediate approach is justified in which only the large eddies are resolved in the simulation whereas the fine grained turbulence is modeled. The LES can be viewed as a hybrid approach using partial simulation (large eddies) and partial modeling (small eddies).

Scale Resolving The alternative to RANS models are models which resolve at least a portion of the turbulence for at least a portion of the flow domain. Such models are generally termed ‘Scale-Resolving’.

rationale behind LES Momentum, mass, energy, and other passive scalars are transported mostly by largeeddies. Large eddies are more problem-dependent. They are dictated by the geometries and boundary conditions of the flow involved.

Small eddies are less dependent on the geometry, tend to be more isotropic, and are consequently more universal. The chance of finding a universal turbulence model is much higher for small eddies.

Resolving only the large eddies allows one to use much coarser mesh and larger times step sizes in LES than in DNS. However, LES still requires substantially finer meshes than those typically used for RANS calculations. In addition, LES has to be run for a sufficiently long flow-time to obtain stable statistics of the fl ow being modeled. As a result, the computational cost involved with LES is normally orders of magnitudes higher than that for steady RANS calculations in terms of memory (RAM) and CPU time. Therefore, high-performance computing (e.g., parallel computing) is a necessity for LES, especially for industrial applications.

Filtered N-S The governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in either Fourier (wave-number) space or configuration (physical) space. The filtering process effectively filters out the eddies whose scales are smaller than the filter width or grid spacing used in the computations.

The resulting equations thus govern the dynamics of large eddies. domain Filtered variable Filter function

The finite volume disretization in FLUENT Volume of the computational cell

Filtering the N-S

Subgrid - scale stress Stress tensor due to molecular viscosity

The subgrid-scale stresses resulting from the ltering operation are unknown, and require modeling. The subgrid-scale turbulence models in FLUENTemploy the Boussinesq hypothesis as in the RANS models, computing subgrid-scale turbulent stresses from Subgrid-scale turbulent viscosity Rate of strain tensor

Smagorinsky-Lilly model, dynamic Smagorinsky-Lilly model, WALE model, dynamic kinetic energy subgrid-scale model.

Smagorinsky-Lilly model Mixing length for subgrid scale Von Karman constant Distance to the closest wall Smog. Const. Volume of the comp. cell

Lilly derived a value of 0.17 for C s for homogeneous isotropic turbulence in the inertial subrange. However, this value was found to cause excessive damping of large-scale fluctuations in the presence of mean shear and in transitional flows as near solid boundary, and has to be reduced in such regions.

In short, Cs is not an universal constant, which is the most serious shortcoming of this simple model. Cs value of around0.1 has been found to yield the best results for a wide range of flows, and is the default value in FLUENT. 0.1 has been found to yield the best results for a wide range of flows, and is the default value in FLUENT. In dynamic Smagorinsky-Lilly model,Cs, is dynamically computed based on the information provided by the resolved scales of motion.

The Wall-Adapting Local Eddy- Viscosity (WALE) Model WALE model is designed to return the correct wall asymptotic behavior for wall bounded flows. C w = 0.325

The Dynamic Kinetic Energy Subgrid- Scale Model The original and dynamic Smagorinsky-Lilly models are essentially algebraic models in which subgrid- scale stresses are parameterized using the resolved velocity scales. The underlying assumption is the local equilibrium between the transferred energy through the grid-filter scale and the dissipation of kinetic energy at small subgrid scales. The subgrid-scale turbulence can be better modeled by accounting for the transport of the subgrid-scale turbulence kinetic energy.

The subgridscale kinetic energy obtained by contracting the subgrid-scale stress in Equation

Filter size

Transport equation for k sgs the model constants, C k and C ε are determined dynamically. σ k =1.0

Filters

In homogenous turbulence

Properties of Filters

Examples of Filters The Top Hat filter This is also called the Box filter

The Gaussian filter This filter is smooth and progressive and thus will include a part (although very small) of the small eddies into the definition of resolved filtered scales.

The structure of strongly non-homogenous flows often requires us to use very different meshes in the three directions in space, thus bringing anisotropies in the three-dimensional mesh and consequently the characteristic length scale of subgrid scale turbulence (often used in closure models) is more difficult to approximate.

Filtering due to the discretization mesh Discretizing on a grid with a step size larger than the Kolmogorov scale will smooth out the higher order modes and consequently imply a filtering operation whose properties are generally not well known.

There are two types of filtering filtering operation is explicitly made (pre-filtering) the explicit filter must have a width larger than the calculation step size and the closure model accounting for the non-resolved scales will be viewed as a subfilter-scale model (SFS). the numerical discretization grid also plays the role of the filter. the grid cell acts both as a numerical discretization cell and as a filter, the model is then properly a subgridscale model (SGS). The discretization and the filtering schemes are connected.