“Big whorls have little whorls that feed on their velocity, and little whorls have lesser whorls and so on to viscosity” – Lewis Fry Richardson
The above quote, an almost epic description by Lewis Fry Richardson captures the horrific cartoon presented by Navier-Stokes equations. Although seemingly a fairly simple set of equations, analytic solutions to even the most simple turbulent flows is out of reach, and one must suffice in obtaining a solution for the flow variables only by numerical confrontation with the equations.
Even though we live in remarkable times (or in other words obey “Moore’s Law”… 😉 ) and emerging hardware such as multiple core CPUs and GPUs are ever so common today, it seems that numerically tackling the Navier-Stokes equations head on – Direct Numerical Simulation (DNS) is prohibitive for most engineering applications of practical interest and it shall remain so for the foreseeable future, its use confined to simple geometries and a limited range of Reynolds numbers in the aim of supplying significant insight into turbulence physics that can not be attained in the laboratory.
Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds Averaged Simulation (RANS), the true “working horse” of modern industrial oriented CFD is based on the Reynolds decomposition according to which a flow variable is decomposed into mean and fluctuating quantities. When the decomposition is applied to Navier-Stokes equation an extra term known as the Reynolds Stress Tensor arises and a modelling methodology is needed to close the equations. The “closure problem” is apparent as higher and higher moments of the equations are extracted, more unknown terms arise and the number of equations never suffices – the closure problem deepens. This is of course an obvious consequence to the fact that taking these higher moments is simply a mathematical endeavour and has no physical addition whats so ever.
I shall address the closure problem in the following paragraphs as it shall reemerge as modeling the “last unresolved problem of classic physics” – turbulence, shall take on sophisticated turns.
As RANS has shown poor performance due its inherent limitation when applied to flows of which strong instabilities and large unsteadiness occurs and it does not seem that a breakthrough in achieving a universal modeling methodology is expected soon (or at all…), researchers have shifted much of the attention and effort to Large Eddy Simulation (LES).
In LES the large energetic scales are resolved while the effect of the small unresolved scales is modeled using a subgrid-scale (SGS) model and tuned for the generally universal character of these scales. LES has severe limitations in the near wall regions, as the computational effort required to reliably model the innermost portion of the boundary layer (sometimes constituting more than 90% of the mesh) where turbulence length scale becomes very small is far from the resources available to the industry. Anecdotally, best estimates speculate that a full LES simulation for a complete airborne vehicle at a reasonably high Reynolds number will not be possible until approximately 2050. Nonetheless, for highly unsteady, vortex dominating flows of which the physical phenomena is mainly derived by the large eddies, LES might be affordable and prevails.
The first, simplest and most widespread sub-grid scale model is the Smagorinsky model, an eddy viscosity model relating the sub-grid scale stresses to the local flow mean strain through the Boussinesq hypothesis.
Smagorinsky subgrid-scale eddy-viscosity model
As much as the model is straightforward it has some inherent deficiencies. Among is the fact that the Smagorinsky constant is always positive and uniform and as a direct consequence so is the eddy viscosity, which is especially problematic for sheared laminar flows, rendering it as inherently unfit for the prediction of laminar to turbulent flow transition.
Another somewhat related deficiency is found in the effect of wall proximity and the value of the eddy viscosity in the near wall region, as the model requires the aid of ad-hoc damping functions and by such the eddy viscosity behavior which should be of the order of the third power of the wall normal proximity is far from attained.
Moreover, Smagorinsky model cannot account for turbulent energy flow from small scales to large scales, which at least locally coud be significant (Backscatter).
My previous post: That’s a Big W(h)ALE… reflected on how the specific shortcomings presented above may be alleviated under simple scrutiny of the physics at hand.
The Dynamic Model (M. Germano et. al.)
In the original dynamic model as proposed by Germano, indirectly relying on scale-similarity concepts, another filter (Test filter), dependable on the grid filter is added. This operation is equivalent to drawing information about the unknown subgrid scales from the smallest resolved scales.
As the test filter width is closer to the grid filter width the resolved turbulent stresses shall be contaminated by a greater amount of numerical error. On the other hand, keeping the ratio between the test filter width and the grid filter too large might imply that large turbulent energy carrying eddies shall be the main contributors in determining the unresolved subgrid scales. Nothing in turbulence modeling is out of subtleties…
Saying all that it is clear that the resolved turbulent stresses intermediate between the grid filter width and that of the test filter are contributing to the Reynolds stresses. The object of the procedure in the original dynamic model is to dynamically propose a local value for the Smagorinsky constant (as a note all relations between the achieved stresses are algebraic)
The Standard Dynamic LES SGS Model (Germano)
although the original aim of the dynamic model was to alleviate the specific deficiencies stated in the above link it is interesting to note:
- Scale-similarity is a feature clearly evident in the inertial subrange for high enough Reynolds number. In as much as the dynamic procedure far and foremost intended to find the smagorinsky constant, it shows no superiority over empirical procedure as the Smagorinsky model is tuned in advance to perform well for such conditions.
- The original dynamic procedure wa most succesful in alleviating the smagorinskys performance for laminar flows, transitional flows and (very) near wall flows – none of which hold the scale-similarity feature.
So why does it actually work?
The main reason is that there is no direct appeal for scale similarity rather, the aim is to select the model constant to depend as little as possible on the level of filtered velocity on which the prediction is based.
The Closure and Completeness Problem of a Turbulence Model.
A turbulence model shall be deemed as complete if its constituent equations are free from flow-dependent a-priori specifications. In as much, although much physical reasoning in construction of the equations, the k-ω SST model as presented in Understanding The k-ω SST Model, is complete.
It should be underlined, that LES models are essentially incomplete. The general practice considers generating a grid characterized by its spacing and to select a filter width determining the physical fidelity, proportional to it, rendering such an important feature – flow dependent and prescribed subjective (in many cases mostly due to lack of computational resources in an unsatisfactory manner).
The dynamic procedure may actually contribute to LES completeness through the concept of “Solution-Adaptive Gridding” or Adaptive LES.
Three important steps were proposed by S B Pope in the quest for grid independent LES:
- Measure the level of the turbulence resolution
- Measure the turbulence resolution length scale as a spatially and temporally
- Specify a turbulence resolution tolerance
In regions where the turbulence resolution exceeds the tolerance resolution tolerance the grid is refined and vice-versa. LES is then performed with adaptive grid. Resolution length scale (filter width) is adjusted through the grid size to match the turbulence resolution tolerance which consequently should be part of the model.
In current practice, LES is incomplete because the turbulence filter width (turbulence resolution scale) is determined subjectively. The variation of the filter width (by variation of the grid) such that the measure of the level of turbulence resolution (1) shall not exceed a resolution tolerance shall render it complete.
The main role of such a procedure is to have the turbulence statistics, which are the best measurement for the quality of the LES by minimizing their dependence on the resolution filter width.
While the implementation of such a concept shall not be an easy task, it is certainly a challenge for future infrastructures of turbulence modeling such as F. Menter Stress Blending Simulation, as it serves more as a modular infrastructure for incorporating (any) RANS with (any) LES into hybrid simulations.
Going Dynamic II to be posted soon shall regard some in-depth concepts of dynamic LES SGS modelling advancements and some shortcomings and pitfalls…