Second Generation URANS – Get Your LES-Like Content Resourcefully

The post shall present a fascinating route to achieving resourceful Scale Resolving Simulation (SRS – P. Moin, F. Menter, P.G Tucker, NASA Report by J. Slotnick, etc’…), all of which aim to provide a road Second Generation URANS – extracting your LES-like content (along with TENZOR insight and conclusions on the subject).

Scale-Resolving Simulations (SRS)

Computational Fluid Dynamics (CFD) progress has been tremendous in the past half a decade.

Moore’s Law vision of an exponential growth in computational resources lived up to its expectation and it’s predicted to keep doing so (at least) for  the next 20 years.

Moore’s Law applied to CFD

Scientists and engineers have developed models of many levels of ﬁdelity for ﬂowﬁelds. On the ladder of CFD one may find many stages. Lifting-Surface Methods that model only the camber lines of lifting surfaces, not the thickness, vortex wakes that must of course be paneled. Linear Panel Methods that solve either the incompressible potential-flow equation or one of the versions applicable to compressible flow with small disturbances. Nonlinear Potential Methods where the velocity is represented as the gradient of a potential, as it is in incompressible potential flow, nonlinearity through effectively incorporating an entropic relation for the density as a function of the local Mach number. Euler Methods, solving the Navier-Stokes equations with the viscus and heat-conduction terms omitted. Coupled Viscous/Inviscid Methods solving the boundary-layer equations in the inner near wall region and matched to an outer region inviscid flow calculations.

One huge leap forward was achieved through the ability to simulate Navier-Stokes Methods Such as Reynolds-Averged Navier-Stokes (RANS).

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.

Reynolds-Stress Tensor

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.

A drawback evident in almost all eddy-viscosity models is the inability to inherently account for rotation and curvature. This drawback is resulted from relating the Reynolds stress to the mean flow strain and in fact is the major difference between such a modeling approach and a full Reynolds-stress model (RSM). The RSM approach accounts for the important effect of the transport of the principal turbulent shear-stress. On the other hand, RSM simulations are not computationally cost-effective, in as much that one does get an improved physical fidelity that is worth the time and computational resources consumed, not only that, they often do not converge.

RANS methodology strength has proven itself for wall bounded attached flows due to calibration according to the law-of-the-wall. For free shear flows however, especially those featuring a high level of unsteadiness and massive separation RANS has shown poor performance following its inherent limitations due to the fact that it’s a one-point closure and by that do not  incorporate the effect of strong non-local effects and of long correlation distances characterizing many types of flows of engineering importance.

Alleviation of the “one-point closure” issue, still under RANS framework, are found in second generation URANS Scale-Adaptive Simulation (SAS – F. Menter) and Partially-Averaged Navier-Stokes (PANS – S. Girimaji) turbulence models (a thorough evaluation of the models appears in the links).

Second Generation URANS – SAS and PANS – An Alternative to LES:

Scale Adaptive Simulation (ANSYS Menter and Egorov)

An interesting methodology to simulate Large-Eddy Simulation (LES) like unsteadiness, lies in the midst of RANS and LES and is especially attractive for flows of which strong instabilities of the flow exist, is termed Scale Adaptive Simulation (SAS) (Menter and Egorov, also available in the Fluent code).

k-w SST Vs. SAS

In SAS formulation, two additional transport equations are solved for. The first is the turbulence kinetic energy transport equation (k) and the second for the square root of KL transport equation (hence the name kskl turbulence model).
What distinguishes the KSKL model from other 2-equation closures is the fact that in the last, the turbulence length scale (which may be defined on dimensional grounds by the transported variables) will always approach the thickness of the shear layer, while for KSKL model, the behavior is such that it allows the identification of the turbulent scales from the source terms of the KSKL model to a measure of both the thickness of the shear layer but also for non-homogenous conditions, as the Von-Karman length scale is related to the strain-rate, individual vortices have locally different time constants (inversely to turnover frequencies) and therefore from a certain size dependable upon the local strain rate, they may not be merged to a larger vortex.

Meaning that the Von-Karman length scale gives a first order estimation for the spatial variation.

SST-URANS Vs. SAS –  Circular cylinder in a cross flow at Re=3.6⋅106
( Iso-surface of Q=S2-Ω2, coloured according to the eddy viscosity ratio).

TENZOR (an authorized partner of ANSYS) does frequent use with this model as well as with the Local Correlation-Based Transition Model (LCTM) (Along with dedicated courses to correctly apply these turbulence models) which are unique to the ANSYS solver. The results validation results seem to endure a tremendous improvement where these unique characteristic behaviour of the flow occurs.

The Partially-Averaged Navier-Stokes (PANS) Method

In PANS method, the so-called “partial averaging” concept is invoked, which corresponds
to a filtering operation for a portion of the fluctuating scales. This concept is based on the observation that the optimum resolved-to-modeled ratio will change from one engineering application to another depending on the reciprocal relations between the level of physical fidelity intended, geometry at hand and computational resources available.

The most important feature which is in the foundation of the approach is the averaging-invariance property of Navier-Stokes equation which amounts to the fact that for any resolved-to-modeled ratio achieved by filtering (i.e. partial filtering), the sub-filter scale stress has the same characteristics as the Reynolds stress, therefore similar closure strategies as for RANS may be employed.
This is a very attractive feature since RANS closure strategies are very mature and well-tested as RANS has truly been the work horse for most large-scale engineering applications, in contrast with LES closures which are mostly algebraic and suffer from lack of complex engineering applications validity.
The original PANS model is therefore based on the 2-equation RANS modelling concept and solves two evolution equations for the unresolved kinetic energy and dissipation.

LES Vs. PANS

It is widely known and goes all the way back to Richardson and granted a more precise view by Kolmogorov, that in turbulence physics, large scales contain most of the kinetic energy and much of the dissipation occurs in the smallest scales,  The smaller the unresolved kinetic energy is, the smaller is the modeled-to-resolved ratio and the greater are both computational effort and physical fidelity for a suited numerical resolution. moreover, the highest value that could be attained for the unresolved dissipation implies that RANS and PANS unresolved scales are the same.

The end result for the evolution equations (different coefficients and parameters definitions may be found at S. Girimaji 2005)

The PANS methodology has some very attractive features:

• The PANS methodology is based on the kinetic energy content and the RANS 2-equation closure methodology rather than on a grid-dependent filter, rendering the model as closed in contrast to LES which is essentially an unclosed method.
Perhaps new advances on the route to “grid-independent” LES modelling (S. Pope, U. Piomelli) shall resolve some of the issues but it shall take some time before such methodologies shall find their way to general purpose CFD codes as most of the exploit dynamic LES non-local concepts.
• As the sub-grid scale filter is independent on the grid resolution explicitly but on the unresolved kinetic energy and dissipation there is a decoupling between the physical and numerical resolution.
• The two evolving parameters unresolved kinetic energy and dissipation may be either constant (as a fraction of RANS) or spatial and time dependent (such as in DES) rendering PANS as more of an infrastructure for resolved-scale simulation rather than a simple modelling approach.

PANS simulation of rotationally oscillating bridge segment (by Chalmers University)

The use of PANS in zonal hybrid RANS-LES (L. Davidson – Chalmers University)

A new advancement in the field of hybrid RANS-LES zonal method is the employment of its attractive features to construct a straightforward hybrid infrastructure.

In the application PANS is applied in the URANS subdomain where the unresolved kinetic energy parameter is unity and a tuned value of 0.4 in the LES subdomain. As stated in former paragraphs the imminent issue is consistently defining the interface RANS-LES layer, and in this modelling approach it is done through the use of the unresolved kinetic energy gradient which gives rise to an additional term in
the momentum equations and the K equation above only in the interface and acts as a forcing term in the momentum equation to create a smooth RANS-LES interface.

To conclude…

Many hybrid RANS/LES which introduces the grid spacing into the turbulence model in order to achieve LES treatment, suffer from the “Modeled Stress Depletion” (MSD) Phenomena related to the switch from RANS to LES on an ambiguous grid setup. In DES for example, 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 (hence the term “Modeled Stress Depletion” or MSD), which in turn shall reduce the skin friction and by that may cause early separation.

This does not occur in SAS as it does not incorporate an explicit dependence on the grid to the turbulence model.

Furthermore, while the ultimate goal in hybrid RANS-LES modeling 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” is the most troublesome resolve.
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 too large to replace all the turbulent stresses for the upcoming RANS state.
The end result is often contamination of the LES region due to inconsistent treating of the turbulent stresses in the interface. The “grey area” (A dedicated post shall soon be written 😉 ) is indeed one of the most important issues to be resolved as far as RANS-LES hybrid methods are concerned.
Recent proposals in the field of zonal hybrid RANS-LES include the incorporation of the SAS/PANS models both to supply unsteady content for the RANS-LES interface and performed as frozen simulation in the LES zones to serve for the purpose of a smooth switching at LES-RANS interface, as the SAS/PANS models shall essentially perform as RANS on coarser grids.