“The whole trick is to be able to convert between RANS areas and LES intelligently — and on the fly…” – Florian Menter.
Today’s industry need for rapid answers dictates CFD simulations to be mainly conducted by RANS simulations whose strength has proven itself for wall bounded attached flows due to calibration according to the law-of-the-wall. However, for free shear flows, especially those featuring a high level of unsteadiness and massive separation RANS has shown poor performance following its inherent limitations.
RANS 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 set of equations may be taken, more unknown terms arise and the number of equations never suffices.
Levels of RANS turbulence modelling are related to the number of differential equations added to Reynolds Averaged Navier-Stokes equations in order to “close” them.
0-equation (algebraic) models are the simplest form of turbulence models, a turbulence length scale is specified in advance through experimenting. 0-equations models are very limited in applications as they fail to take into account history effects, assuming turbulence is dissipated where it’s generated, a direct consequence of their algebraic nature.
1-equation and 2-equations models, incorporate a differential transport equation for the turbulent velocity scale (or the related the turbulent kinetic energy) and in the case of 2-equation models another transport equation for the length scale, subsequently invoking the “Boussinesq Hypothesis” relating an eddy-viscosity analog to its kinetic gasses theory derived counterpart (albeit flow dependent and not a flow property) and relating it to the Reynolds stress through the mean strain.
In this sense 2-equation models can be viewed as “closed” because unlike 0-equation and 1-equation models (with exception maybe of 1-equations transport for the eddy viscosity itself) these models possess sufficient equations for constructing the eddy viscosity with no direct use for experimental results.
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…
LES simulation of isotropic turbulence
On the other hand, for free shear flows of which the large eddies are at the order of magnitude as the shear layer, LES may provide extremely reliable information as it’s much easier to resolve the large turbulence eddies in a fair computational effort.
As such, researchers have shifted much of the attention and effort to hybrid formulations incorporating RANS and LES in certain ways. In most hybrid RANS-LES methods RANS is applied for a portion of the boundary layer and large eddies are resolved away from these regions by an LES.
“The Grey Area” – Interfacing RANS and LES
While the ultimate goal is a model that may work in the RANS limit, LES limit and smoothly connect them at their interface (might it be zonal or monolithic formulation), it seems that in particular the interface termed “the grey area” stands problematic although in the focus of the CFD community for some time.
The main reason for that is in the fact that although seemingly the same form of formulation for the governing filtered equation is achieved the nature their derivation and their simulation objectives are fundamentally very different.
The RANS equations assume that a time average is much greater than the turbulent eddies time scale, hence turbulent stresses may be replaced by their averaged effect. usually this is done by defining an eddy viscosity (see Understanding The k-ω SST Model) proportional to the mean strain rate and resulting in a flow that is computationally very stable even at highly turbulent unsteady regions as the effective viscosity can be of orders of magnitude larger the molecular viscosity.
On the other hand, in an LES the formulation is derived by spatial filtering separating the scales that can be directly calculated from those that must be modeled (due to grid resolution – “filter width”). Generally the subgrid scales are also replaced with an effective viscosity that must be low enough as to not artificially damp the growth and transport of the resolved large-scale eddies that are supposed be captured.
In the Interface region the modelled turbulent stresses formerly derived by RANS may easily be too large to maintain those unsteady features desired to be captured by LES, and on the other hand not large enough to replace all the turbulent stresses for the upcoming RANS state.
The end result is seldom contamination of the LES region due to inconsistent treating of the turbulent stresses at the interface. The “grey area” is indeed one of the most important issues to be resolved as far as RANS-LES hybrid methods are concerned.
Detached Eddy Simulation (DES)
One of the most popular hybrid RANS-LES models is Detached Eddy Simulation (DES) devised originally by Philippe Spalart. The term DES is based on the Idea of covering the boundary layer by RANS model and switching the model to LES mode in detached regions thereby cutting the computational cost significantly yet still offering some of the advantages of an LES method in separated regions.
The formulation of the hybridization of the model is fairly straight forward:
This means that as Δ is max(ΔX, ΔY, ΔZ) this modification of the S-A model, changes the interpretation of the model as the modified distance function causes the model to behave as a RANS model in regions close to walls, and as an eddy-viscosity based LES (Smagorinsky, Wale, etc’…) manner away from the walls.
Subtleties in DES formulation
Being so popular, I found some of the natural DES (P. Spalart 1997) inherent limitations are often overlooked in simulations as practitioners often apply the model in order to increase physics fidelity without dwelling on subtle issues. The following paragraphs address some of these subtleties.
In DES the hybrid formulation has a limiter switching from RANS to LES as the grid is reduced. The problem with natural DES is that an incorrect behavior may be encountered for flows with thick boundary layers or shallow separations. It was found that when the stream-wise grid spacing becomes less than the boundary layer thickness the grid may be fine enough for the DES length scale to switch the DES to its LES mode without proper “LES content”, i.e. resolved stresses are too weak (“Modeled Stress Depletion” or MSD”), which in turn shall reduce the skin friction and by that may cause early separation. The phenomenon is termed Grid Induced Separation (GIS).
mean velocity in different ypes of grids in a boundary layer –
top: natural DES, left: ambiguous grid spacing, right: LES
As a consequence of the original DES deficiencies an advancement to the model was devised, termed Delayed-DES (DDES). In the Fluent DES-SST formulation a DES limiter “shield” is added to maintain RANS behavior in the boundary layer without grid dependency.
In subsequent improvements to the DDES formulation, RANS are applied to the innermost portion of the boundary layer and large eddies are resolved away from these regions. In such formulation LES is confined to the rest of the boundary layer or to regions where flow is detached which provides a Wall-Modelled Large-Eddy Simulation (WMLES) of attached flows at high but fair computational cost.
Improved-DDES for the flow behind a circular cylinder
Another subtlety concerns that concerns the “grey area”, specifically the region of transition between RANS and LES models. DES utilizes a model parameter very similar to the one in Smagorinsky LES model which is found deficient in the ability to handle laminar-turbulent transition (among other deficiencies). The same is observed in DES as high levels of eddy viscosity attenuate the transition process which contribute to the “grey area” problem, specifically the RANS to LES transition by interfering with “turbulence content” arising from shear layer instability. This is an ongoing issue with DES and some options to overcome this “grey area” phenomena incorporating local formulation (so as they can be straightforwardly implemented in an OpenFOAM code) have been proposed such as processing the local velocity gradient to distinguish between situations of which the eddy viscosity is low (such as plane shear) to regular turbulence, where the subgrid-scale model of the LES can be in use.
The “grey area” is a common problem for other hybrid formulations as well and there is still a long way to go in that vector…
Delayed Detached-Eddy Simulation (DDES) Formulation
The main corner stone for the DDES hybrid RANS-LES model is the Spalart-Allmaras Turbulence Model. One transport equations for the eddy-viscosity based models such as Spalart-Allmaras don’t have an internal length scale as far as a measyre of the mean shear rate is concerned, but do incorporate a ratio (squared) of a model length scale to the wall distance. The parameter is modified in the DDES formulation to support any eddy viscosity based model (a straightforward procedure to extract an eddy viscosity transport model from a two transport equations model )
where νt is the kinematic eddy viscosity, ν the molecular viscosity, Ui,j the velocity gradients, κ the Kármán constant and d the distance to the wall.
As the length scale is 1 in the logarithmic layer and gradually goes to zero in the boundary layer edge the kinematic viscosity is added to the formulation to ensure its stays correct in high proximity to the wall such that the length scale remains away from zero (exceeding 1).
A function is defined to ensure that the solution will be a RANS solution even if the grid spacing is smaller than the boundary layer thickness (so it will be 1 in the LES region where the length scale defined above is much smaller than 1, and 0 elsewhere while not sensitive in situations of high proximity to the wall when the length scale exceeds 1.
Now an alteration to the DES length scale is proposed such that under specific coefficient values (which the above function is not so sensitive to even in the case of a different formulation of DES other than spalart-Allmaras, say the k-ω SST Model )
In this formulation, when the function is 0, the length scale dictates RANS mode to operate, and when the function is 1 natural DES (P. Spalart 1997) applies. The difference lies in the fact that on contrary to natural DES formulation where the length scale depends solely on the grid, in the DDES formulation it depends also on the eddy-viscosity. This means that the revised formulation will “insists” upon remaining on RANS mode if the grid is inside the boundary layer and if massive separation is encountered, the functions value will switch to LES mode a much more abrupt manner than the switch in the natural DES formulation, rendering the “grey area” narrower which is highly desirable.
The original DDES is set to Spalart-Allmaras eddy-viscosity transport equation to achieve an eddy viscosity (see the link for an in-depth evaluation of the turbulence model) for RANS mode and an eddy-viscosity based LES model (such as WALE for example).
Vorticity isosurfaces in a circular cylinder simulation (F. Spalart 2009)
Eventually both DES and DDES shall perform well for flows of which large instabilities and massive separation occurs (such as the flow behind a cylinder) but the former may prove problematic for thick boundary layers or for flow with weak instabilities. The Improved DDES should be chosen only for flows where LES content of the boundary layer is of high priority (a problem dependent question) otherwise the computational cost shall rise sharply.
All of the above (and more…) reveals that while some envision DES as becoming an everyday design tool, it shall probably remain such only for well experienced LES practitioners at least in the near future…
Now go and IDDES/DDES/DES but watch out… 😉