“…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
Spatial and Temporal Resolution
In LES, especially for complex flows as the large-scale anisotropic features are to be accurately (as much as numerically possible) captured. SGS modeling should apply to structures in the inertial range. This posses again a problem in as much as resolution of near wall region is concerned, the small eddies carry a great amount of turbulent kinetic energy and the “spectral gap problem” renders no scale separation. The end result is that the near wall resolution of LES compares with that of DNS.
Mostly LES should be practiced using high-order spatial numerical methods. I shall leave the elaborate discussion on the “why and what” of the subject to my previous post High Vs. Low Order Numerical Methods in CFD.
Due the obvious nature of LES, mostly conducted to exploit most computational resources, standard grid convergence procedure (convergence to DNS…) is not an option. Instead, a solution on a coarser grid may be obtained. halving the spatial resolution in all directions (keeping the time step constant) shall reduce the computational expense by 1/8. As results should not be expected to be quantitatively comparable to the finer grids results, the physical phenomena must still be apparent as an under-resolved simulation. In case it’s lost, then probably the original mesh resolution should not have been considered as proper LES resolution.
As for temporal resolution, if higher-order spatial numerical schemes and finer grid resolutions are imposed to capture smaller turbulent structures, the time scale should also be adjusted in order to accurately describe the advection and convection of these structures.
Trying to cut the time step to achieve a better temporal view would of course increase the computational resources demand. Trying to do so on a coarser mesh would mean fine scale structures would not be captured, leaving the practitioner with no gain in as much as time step checkups are concerned. In essence time step choosing is a compromise.
Is it possible to have a perfect LES such that the solution to LES equations shall be equal to spatially filtered value of the true velocity field?… Actually no.
As the spatially filtered value of the true velocity field is a random field, its future evolution is not determined by its current state. So even if at some initial moment the LES solution equals the spatially filtered value of the true velocity field, the last has just a statistical distribution and hence no specific value for the solution of the LES equation equals the spatially filtered value of the true velocity field. Meaning the relation between the two entities is statistical.
The immediate consequence is that a-priori testing comparing LES quantities obtained from a specific realization of the true velocity field do not serve as a validation procedure. The procedure should have a set of equations for the resolved velocity field and a measure for some statistics as temporal and wave number spectra (entirely not an easy task…). This could be done by periodically recording complete data sets (memory issues shall arise) and by that obtain statistical information to achieve ensemble averages.
The ultimate validation is achieved by incorporating the LES solution with statistical measurements extracted by the previous described procedure (or some other) in order to get estimates on the statistics of the true turbulent velocity field.
Often, only one-point and first-order statistics are drawn for the assessment of simulations which are unsteady in nature. Considering the fact that in LES has the capability to predict the fluctuating field in complex configurations numerous levels of validation could be achieved. one proposition for the classification of the validation levels is as follows (P. Sagaut and S. Deck 2009):
- Level 1: comparison of integral quantities (e.g. drag or lift).
- Level 2: reproduction of first order statistics (e.g. mean velocity profile).
- Level 3: reproduction of second order statistics (e.g. R.M.S. profile)
- Level 4: one-point spectral analysis (e.g. PSD)
- Level 5: two-point spectral analysis (e.g. correlations, cross-correlations, phase spectra, coherence)
- Level 6: nonlinear coupling and time-frequency analysis (for non-stationary processes).
It should be also stressed that since LES and experiment datasets differ by an orders of magnitude in duration, as the first is most often oversampled on a short duration due to the limitation in computational resources, especially for industrial applications of which complex geometries incorporating wall presence dominating flows. From a statistical standpoint, the problem of drawing statistics for two sequences of data whose duration differs significantly (e.g. CFD and experiments datasets) should be approached with caution.
A snapshot of Large Eddy Simulation of a 5-bladed rotor wake in hover with a novel multiblock IBM
(by Technion CFD Lab)
Following the anticipated approach for LES simulation setup short review 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, continuing with LET’s LES IV, shall review some of the subtleties.