“…the smallest eddies are almost numberless, and large things are rotated only by large eddies and not by small ones, and small things are turned by small eddies and large” – Leonardo di ser Piero da Vinci
Following the anticipated approach for LES simulation setup, as presented thoroughly in LET’s LES I, LET’s LES II and LET’s LES III and assuming one has followed these steps and had the computational resources to actually solve the simulation, pops up the issue of qualifying the filtering applied, interpretation of LES results and the problems entangled in realizing a good enough interpretation of them. The following posts shall address just that.
The Effective Filter Concept:
Here I shall restrict myself to subgrid viscosity models for the ease of explaining the concept. Nonetheless, the concept may intuitively be extended to other type of models which mathematical representation makes it less possible.
subgrid viscosity models use an intrinsic length scale denoted Δf, which can be interpreted as the “mixing length” associated with the subgrid scales (in much of a similar manner as hypothesized by the “Bousinesq Hypothesis”) basing it on the large scale average characteristics of the flow:
By that, a ratio between this mixing length and the filter cutoff length may be established to be recognizable as the subgrid scale constant:
by varying it a modification to the ratio between filter cutoff length and the length scale included in the model is achieved, all the while rendering them as independent.
We may note that the subgrid scales, represented only by the subgrid models which by their effects impose the filter on the computed solution. Since the last are only a model for the exact solution by the computed one, applying such a filter only models the the imposition of the formal filter.
Such a transfer may be ensured by applying an implicit filter, which is contained in each subgrid model. Such a dynamic, implicit concept of the filtering process takes the modeling errors into account and therefore the process may be described as twofold:
- The level of representation of the physical system which is represented by application of a filter using the Navier–Stokes equations (NSE) as a convolution product.
- The existence of a cutoff length in the subgrid in use.
Applying both filters sums up in an important concept which is the realization of the filtering by an effective filter which is the one actually perceived by the system. To construct such a filter, a general grasp of the contribution for each of the operation should be established.
For large values of the Smagorinsky constant (Cs > 0.5), the characteristic cutoff length is the mixing length produced by the model. The model then dissipates more energy than if it were actually located at the scale Δ because it ensures the energy flux balance through the cutoff associated with a longer characteristic length. The effective filter is therefore fully determined by the subgrid model. This solution criterion should be compared with the one defined for hot-wire measurements, which recommends that the
wire length be less than twice the Kolmogorov scale in developed turbulent
For small values of the Smagorinsky constant (Cs < 0.15), it is the cutoff length Δ that plays the role of characteristic length and the effective filter corresponds to the usual
analytical filter. It should be noted in this case that the energy drainage induced by the model is less than the transfer of kinetic energy through the cutoff, so the energy balance is no longer maintained. This is reflected in an accumulation of energy in the resolved scales, and the pertinence of the simulation results should be taken with caution.
For intermediate values of the Smagorinsky constant (Cs ≈ 0.2), the effective filter is a combination of the analytical filter and model’s implicit filter, which makes it difficult to
interpret the dynamics of the smallest resolved scales. The dissipation induced
by the model in this case correctly insures the equilibrium of the energy fluxes
through the cutoff.
To conclude, a complete analysis for the constants value presented is not in the scope of the post, nonetheless it is valuable but may be used only for reference, because it is based on very restrictive hypotheses (an eddy viscosity Bousinesq like hypotheses). Nevertheless it indicates that the numerical error is not negligible and that it can even be dominant in certain cases over the subgrid terms. The effective numerical filter is then dominant over the scale separation filter and this should be a great acknowledgement do the inconsistency of ILES methodology that shall be discussed in the near future of LET’s LES following posts.
This error can be reduced either by increasing the order of accuracy of the numerical scheme or by using a pre-filtering technique that decouples the cutoff length of the analytical filter of the discretization step (Ghosal et al.).