### 12.1.5. Large Eddy Simulation (LES)

The details of different LES models can be found in the User and Theory documentation of the corresponding solvers. As described in Section 12.1.5.1, the main purpose of LES models is to provide sufficient damping for the smallest (unresolved) scales. For this reason, it is not advisable to use complex formulations, but to stay with simple algebraic models. The most widely used LES model is the Smagorinsky, 1963 [29] model:

| (12–13)

The main deficiency of the Smagorinsky model is that its eddy-viscosity does not go to zero for laminar shear flows (only ). For this reason, this model also requires a near-wall damping function in the viscous sublayer. It is desirable to have a LES formulation that automatically provides zero eddy-viscosity for simple laminar shear flows. This is especially important when computing flows with laminar turbulent transition, where the Smagorinsky model would negatively affect the laminar flow. The simplest model to provide this functionality is the WALE (Wall-Adapting Local Eddy-viscosity) model of Nicoud and Ducros, 1999 [22]. The same effect is also achieved by dynamic LES models, but at the cost of a somewhat higher complexity. None of the classical LES models addresses the main industrial problem of excessive computing costs for wall-bounded flows at moderate to high Reynolds numbers.

There are numerous cases at very low Reynolds numbers where LES can be an industrial option. Under such conditions, the wall boundary layers are likely laminar and turbulence forms only in separated shear layers and detached flow regions. Such situations can be identified by analyzing RANS eddy viscosity solutions for a given flow. In the case where the ratio of turbulence to molecular viscosity is smaller than inside the boundary layer, it can be assumed that the boundary layers are laminar and no resolution of near-wall turbulence is required. Such conditions are observed for flows around valves or other small-scale devices at low Reynolds numbers.

LES can also be applied to free shear flows, where resolution requirements are much reduced relative to wall-bounded flows.

#### 12.1.5.1. Limitations of Large Eddy Simulation (LES)

In order to understand the motivation for hybrid models, one has to discuss the limitations of Large Eddy Simulation (LES). LES has been the most widely used SRS model over the last decades. It is based on the concept of resolving only the large scales of turbulence and to model the small scales. The classical motivation for LES is that the large scales are problem-dependent and difficult to model, whereas the smaller scales become more and more universal and isotropic and can be modeled more easily.

LES is based on filtering the Navier-Stokes equations over a finite spatial region (typically the grid volume) and aimed at only resolving the portions of turbulence larger than the filter width. Turbulence structures smaller than the filter are then modeled, typically by a simple Eddy Viscosity model.

The filtering operation is defined as:

| (12–14)

where is the spatial filter. Filtering the Navier-Stokes equations results in the following form (density fluctuations neglected):

| (12–15)

The equations feature an additional stress term due to the filtering operation:

| (12–16)

Despite the difference in derivation, the additional sub-grid stress tensor is typically modeled as in RANS using an eddy viscosity model:

| (12–17)

The important practical implication from this modeling approach is that the modeled momentum equations for RANS and LES are identical if an eddy-viscosity model is used in both cases. The modeled Navier-Stokes equations have no knowledge of their derivation. The only information they obtain from the turbulence model is the level of the eddy viscosity. Depending on that, the equations will operate in RANS or LES mode (or in some intermediate mode). The formal identity of the filtered Navier-Stokes and the RANS equations is the basis of hybrid RANS-LES turbulence models, which can obviously be introduced into the same set of momentum equations. Only the model (and the numerics) have to be switched.

Classical LES models are of the form of the Smagorinsky, 1963 [29] model:

| (12–18)

where is a measure of the grid spacing of the numerical mesh, is the strain rate scalar and is a constant. This is obviously a rather simple formulation, indicating that LES models will not provide a highly accurate representation of the smallest scales. From a practical standpoint, a very detailed modeling might not be required. A more appropriate goal for LES is not to model the impact of the unresolved scales onto the resolved ones, but to model the dissipation of the smallest resolved scales. This can be seen from Figure 12.6 showing the turbulence energy spectrum of a Decaying Isotropic Turbulence (DIT) test case; that is, initially stirred turbulence in a box, decaying over time (Comte-Bellot and Corrsin, 1971 [3]). is the turbulence energy as a function of wave number . Small values represent large eddies and large values represent small eddies. Turbulence moves down the turbulence spectrum from the small wave number to the high wave numbers. In a fully resolved simulation (Direct Numerical Simulation, or DNS), the turbulence is dissipated into heat at the smallest scales ( in Figure 12.6), by viscosity. The dissipation is achieved by:

| (12–19)

where is typically a very small kinematic molecular viscosity. The dissipation is still of finite value as the velocity gradients of the smallest scales are very large.

LES computations are usually performed on numerical grids that are too coarse to resolve the smallest scales. In the current example, the cut-off limit of LES (resolution limit) is at around . The velocity gradients of the smallest resolved scales in LES are therefore much smaller than those at the DNS limit. The molecular viscosity is then not sufficient to provide the correct level of dissipation. In this case, the proper amount of dissipation can be achieved by increasing the viscosity, using an eddy-viscosity:

| (12–20)

The eddy viscosity is calibrated to provide the correct amount of dissipation at the LES grid limit. The effect can be seen in Figure 12.6, where a LES of the DIT case is performed without a LES model and with different LES models. When the LES models are activated, the energy is dissipated and the models provide a sensible spectrum for all resolved scales. LES is not modeling the influence of unresolved small scale turbulence onto the larger, resolved scales, but the dissipation of turbulence into heat (the dissipated energy is typically very small relative to the thermal energy of the fluid and does not have to be accounted for, except for high Mach number flows).

Figure 12.6: Turbulence spectrum for DIT test case after t=2. Comparison of results without Sub-Grid Scale model (no LES) with WALE and Smagorinsky LES model simulations

This discussion shows that LES is a fairly simple technology, which does not provide a reliable backbone for modeling. This is also true for more complex LES models like dynamic models. Dynamic eddy viscosity LES models (see Geurts, 2004 [12]) are designed to estimate the required level of dissipation at the grid limit from flow conditions at larger scales (typically twice the filter width), thereby reducing the need for model calibration. Such models, however, also only provide a suitable eddy viscosity level for energy dissipation. As a result, within the LES framework, all features and effects of the flow that are of interest and relevance to engineers have to be resolved in space and time. This requirement makes LES a very CPU-expensive technology.

Even more demanding is the application of LES to wall-bounded flows, which is the typical situation in engineering flows. The turbulent length scale, , of the large eddies can be expressed as:

| (12–21)

where is the wall distance and a constant. Even the (locally) largest scales become very small near the wall and require a high resolution in all three space dimensions and in time.

The linear dependence of on indicates that the turbulence length scales approach zero near the wall, which would require an infinitely fine grid to resolve them. This is not the case in reality, as the molecular viscosity prevents scales smaller than the Kolmogorov limit. This is manifested by the viscous or laminar sublayer, a region very close to the wall, where turbulence is damped and does not need to be resolved. However, the viscous sublayer thickness is a function of the Reynolds number, Re, of the flow. At higher Re numbers, the viscous sublayer becomes decreasingly thinner and thereby allows the survival of smaller and smaller eddies, which need to be resolved. This is depicted in Figure 12.7 showing a sketch of turbulence structures in the vicinity of the wall (for example, channel flow with flow direction normal to observer). The upper part of the picture represents a low Re number and the lower part a higher Re number situation. The gray box indicates the viscous sublayer for the two Re numbers. The structures inside the viscous sublayer (circles inside the gray box) are depicted but not present in reality due to viscous damping. Only the structures outside of the viscous sublayer (that is, above the gray box) exist and need to be resolved. Due to the reduced thickness of the viscous sublayer in the high Re case, substantially more resolution is required to resolve all active scales. Wall-resolved LES is therefore prohibitively expensive for moderate to high Reynolds numbers. This is the main reason why LES is not suitable for most engineering flows.

Figure 12.7: Sketch of turbulence structures for wall-bounded channel flow with viscous sublayer (a) Low Re number (b) High Re number (Grey area: viscous sublayer)

The Reynolds number dependence of wall-resolved LES can be estimated for a simple periodic channel flow as shown in Figure 12.8 (-streamwise, -wall-normal, -spanwise, is the channel height).

| (12–22)

Figure 12.8: Turbulence structures in a channel flow

The typical resolution requirements for LES are:

| (12–23)

where is the non-dimensional grid spacing in the streamwise direction, in the spanwise and the number of cells across half of the channel height. With the definitions:

| (12–24)

one can find the number, of cells required as a function of , for resolving this limited domain of simple flow (see Table 12.1).

| (12–25)

Table 12.1: Number of cells, , vs Reynolds number for channel flow

| 500 | 103 | 104 | 105

(For the practitioner: the Reynolds, , number based on the bulk velocity is around a factor of ten larger than the Reynolds number, , based on friction velocity. Note that is based on ). The number of cells increases strongly with number, demanding high computing resources even for very simple flows. The CPU power scales even less favorably, as the time step must also be reduced to maintain a constant CFL number ().

The Re number scaling for channel flows could be reduced by the application of wall functions with increasing values for higher numbers. However, wall functions are a strong source of modeling uncertainty and can undermine the overall accuracy of simulations. Furthermore, the experience with RANS models shows that the generation of high quality wall-function grids for complex geometries is a very challenging task. This task is even more challenging for LES applications, where you would have to control the resolution in all three space dimensions to conform to the LES requirements (for example, and then depend on ).

For external flows, there is an additional Re number effect resulting from the relative thickness of the boundary layer (for example, boundary layer thickness relative to chord length of an airfoil). At high Re numbers, the boundary layer becomes very thin relative to the body’s dimensions. Assuming a constant resolution per boundary layer volume, Spalart et al., 1997 [30], 2000 [31] provided estimates of computing power requirements for high Reynolds number aerodynamic flows under the most favorable assumptions. Even then, the computing resources are excessive and will not be met even by optimistic estimates of computing power increases for several decades, except for simple flows.

While the computing requirements for high Re number flows are dominated by the relatively thin boundary layers, the situation for low Re number technical flows is often equally unfavorable, as effects such as laminar-turbulent transition dominate and need to be resolved. Based on reduced geometry simulations of turbomachinery blades (see Michelassi, 2003 [20]), an estimate for a single turbine blade with end-walls is given in Table 12.2.

Table 12.2: Computing power estimate for a single turbomachinery blade with end-walls

Method | Cells | Time steps | Inner loops per time step | Ratio to RANS
--- | --- | --- | --- | ---
RANS | ~106 | ~102 | 1 | 1
LES | ~108–109 | ~104–105 | 10 | 105–107

Considering that the goal of turbomachinery companies is the simulation of entire machines (or at least significant parts of them), it is unrealistic to assume that LES will become a major element of industrial CFD simulations even for such low Re number () applications. However, LES can play a role in the detailed analysis of elements of such flows like cooling holes or active flow control.

All the above does not mean that LES of wall-bounded flows is not feasible at all, but just that the costs of such simulations are high. Figure 12.9 shows the grid used for a LES around a NACA 0012 airfoil using the WALE model. The computational domain is limited in the spanwise direction to 5% of the airfoil chord length using periodic boundary conditions in that direction. At a Reynolds number of a spanwise extent of 5% has been estimated as the minimum domain size that allows turbulence structures to develop without being synchronized across the span by the periodic boundary conditions. The estimate was based on the boundary layer thickness at the trailing edge as obtained from a precursor RANS computation. This boundary layer thickness is about 2% chord length. The grid had 80 cells in the spanwise direction and overall cells. The simulation was carried out at an angle of attack of , using ANSYS Fluent in incompressible mode. The chord length was set to , the freestream velocity, and the fluid is air at standard conditions. The time step was set to giving a Courant number of inside the boundary layer. Figure 12.10 shows turbulence structures near the leading edge (a) and the trailing edge (b). Near the leading edge, the laminar-turbulent transition can clearly be seen. The transition is triggered by a laminar separation bubble. Near the trailing edge, the turbulence structures are already relatively large, but still appear unsynchronized in the spanwise direction (no large scale 2D structures with axis orientation in the spanwise direction). The simulation was run for ~104 time steps before the averaging procedure was started. The time averaging was conducted for time steps. Figure 12.11 shows a comparison of the wall pressure coefficient and Figure 12.12 of the wall shear stress coefficient on the suction side of the airfoil in comparison to a RANS computation using the SST model (Menter, 1994 [16]). No detailed discussion of the simulation is intended here, but the comparison of the wall shear stress with the well-calibrated RANS model indicates that the resolution of the grid is still insufficient for capturing the near-wall details. For this reason, the wall shear stress is significantly underestimated by about 30% compared to the SST model in the leading edge area. As the trailing edge is approached, the comparison improves, mainly because the boundary layer thickness is increased whereas the wall shear stress is decreased, which produces a higher relative resolution in the LES. Based on this simulation, it is estimated that a refinement by a factor of 2, in both streamwise and spanwise directions, would be required in order to reproduce the correct wall shear stress. While such a resolution is not outside the realm of available computers, it is still far too high for day-to-day simulations.

Figure 12.9: Details of grid around a NACA 4412 airfoil (a) Grid topology (b) Leading edge area (c) Trailing edge area

Figure 12.10: Turbulence structures of WALE LES computation around a NACA 4412 airfoil (a) Leading edge (b) Trailing edge (Q-criterion, color- spanwise velocity component)

Figure 12.11: Wall pressure coefficient Cp on the suction side of a NACA 4412 airfoil: comparison of RANS-SST and LES-WALE results

Figure 12.12: Wall shear stress coefficient Cf on the suction side of a NACA 4412 airfoil: comparison of RANS-SST and LES-WALE results

Overall, LES for industrial flows will be restricted in the foreseeable future to flows not involving wall boundary layers, or wall-bounded flows in strongly reduced geometries, preferentially at low Re numbers.

The limitations of the conventional LES approach are the driving force behind the development of hybrid RANS-LES models that are described in the later parts of this report.

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