“The whole trick is to be able to convert between RANS areas and LES intelligently — and on the fly…” – Florian Menter (ANSYS).
Reynolds-Averaged Navier-Stokes (RANS)
Today’s industry need for rapid answers dictates CFD simulations to be mainly conducted by Reynolds-Averaged Navier-Stokes (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.
2-equations models do however contain many assumptions along the way for achieving the final form of the transport equations and as such are calibrated to work well only according to well-known features of the applications they are designed to solve. Nonetheless although their inherent limitations, today industry need for rapid answers dictates CFD simulations to be mainly conducted by 2-equations models whose strength has proven itself for wall bounded attached flows at high Reynolds number (thin boundary layers) due to calibration according to the law-of-the-wall.
Approaching Large-Eddie 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…
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.
The original DES 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).
The actual formulation for a two-equation model is (the turbulence kinetic energy equation of a k-ω model):
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.
Grid Induced Separation
Being so popular, some of the natural DES (P. Spalart 1997) inherent limitations were 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 (following references from P. Spalart et al. 2006 and F. R. Menter 2000).
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 types 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.
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 measure 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 – we shall see such a formulation shortly)
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)
For two-equation models, the dissipation term in the turbulence kinetic energy equation is formulated as follows:
It is worth mentioning that DES and its variants are termed and essentially are global hybrid methods.
Global hybrid methods are based on a continuous treatment of the flow variables at the interface between RANS and LES and by that introduce a ‘grey area’ in which the solution is neither pure RANS nor pure LES since the switch from RANS to LES does not imply an instantaneous change in the resolution level. These methods can be considered as weak RANS–LES coupling methods since there is no mechanism to transfer the modelled turbulence energy into resolved turbulence energy.
In the above formulation The function FDDES is designed as to reach unity inside the wall boundary layer and zero away from the wall. The definition of this function is intricate as it involves a balance between proper shielding and not suppressing the formation of resolved turbulence as the flow separates from the wall. As the function FDDES blends over to the LES formulation near the boundary layer edge, no perfect shielding can be achieved. The limit for DDES is typically in the range of the maximum edge length of the local computational cell is less then 20% of the boundary layer thickness which allows for meshes where the maximum edge length of the local computational cell is of 20% than for natural DES. However, even this limit
is frequently reached so the GIS phenomena is not fully prevented with DDES.
There are a number of DDES models available in ANSYS Fluent/CFX. They follow the same principal idea with respect to switching between RANS and LES mode. The models differ therefore mostly by their RANS capabilities and should be selected accordingly.
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.
Shielded Detached Eddy Simulation (SDES)
The SDES formulation is yet another variation of DES. The improvement is in the shielding function and the interaction with the grid scale. This is emphasized in the turbulence model by an additional sink term in the turbulence kinetic energy equation:
The shielding function in the SDES formulation (namely – fs) provides more shielding then the corresponding shielding function in the DDES formulation (F-DDES), this means that the original shielding based on the mesh length scale can be reduced and is therefore defined in SDES as:
The first part in the above is the conventional LES mesh length scale, the second is again based on the maximum edge length as in the DES formulation and the 0.2 in the above ensures that for highly stretched meshes the grid length scale is a fifth of that of DDES and another implication is the reduction of the eddy-viscosity in LES mode by a factor of 25 as it is dependent quadratically upon the grid size. This is an important artifact as it improves the RANS to LES transition of DES models.
In engineering flows, flow characteristics of shear flows is much more encountered than that of decaying isotropic turbulence (DIT). The last is the basis for the calibration of the DES/DDES constant. Shear flows the Smagorinsky constant is reduced and this is achieved by setting the constant in SDES to 0.4.
Now if we combine the above explained effect of the grid scale on the eddy viscosity with the modified constant a reduction by a factor of nearly 60 is achieved for separated flows on stretched grids which is favorably affects the RANS to LES transition.
A New Paradigm: Stress-Blended Eddy Simulation (SBES)
SBES is not a new hybrid RANS-LES model, but a modular approach to blend existing models to achieve optimal performance. In this sense SBES is a modular approach which allows the CFD practitioner to use a pre-selected RANS and another pre-selected LES model instead of the mix of both formulations within one set of equations.
This becomes handy in certain fields of which the modeling sophistication is to be extended from what was originally practiced with a specific and validated LES to include parts of the domain which can only be covered by RANS models without having to replace the trusted LES.
SBES model concept is built on the SDES formulation. In addition, SBES is using the shielding function to explicitly switch between different turbulence model formulations in RANS and LES mode.
For the general case one of the (RANS or LES) models is not based on the eddy viscosity concept the general formulation is presented either in modeled stress tensor:
For the case where both RANS and LES models are based on the eddy viscosity concepts, the formulation simplifies to:
The strong shielding is important for such a formulation to work in order to maintain a zero pressure gradient RANS boundary layer in any grid.
The intention of the SBES methodology is to resolve the following issues (F. R. Menter 2016):
- Exhibit an asymptotic shielding of the RANS boundary layers.
- perform an explicit switch to user-specified LES model in LES region.
- Allowance of rapid ‘transition’ from RANS to LES regions Allow practitioners to be able to clearly distinguish regions where the models run in RANS and regions where the model runs in LES mode.
- Allow Wall-modeled LES capability once in regions of sufficient numerical resolution and an upstream trigger into LES-mode for WMLES simulations.