“Turbulence is the most important unsolved problem of classical physics…” – Richard Feynman.
Nothing would be as cost-effective for the flight industry as the ability to cut substantially on fuel budget. From a CFD perspective, such a goal could be achieved in a much more straightforward manner if a full airborne vehicle simulation could be supplied, still keeping high fidelity of the physics.
A plea for a direct numerical description of the equations is a mixed blessing as it seems the availability of such a description is directly matched to the power of a dimensionless number reflecting on how well momentum is diffused relative to the flow velocity (in the cross-stream direction) and on the thickness of a boundary layer relative to the body – The Reynolds Number.
It is found that the computational effort in Direct Numerical Simulation (DNS) of the Navier-Stokes equations rises as Reynolds number in the power of 9/4 which renders such calculations as 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.
Having said all that, engineering applications could not have been left out and simplified methodologies to capture flow features of interest were developed, their complexity and range of applicability dictated by the simplifying assumption, a direct consequence of computational effort limitations and generally predicted by “Moore’s Law”.
One huge leap forward was achieved through the ability to simulate Navier-Stokes Methods Such as Reynolds-Averged Navier-Stokes (RANS), Large Eddy Simulation (LES) and hybrid RANS-LES Methods.
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 of 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.
So a full airborne vehicle (very) high fidelity simulation is not an option. what is then?…
Most of nowadays CFD simulations are conducted with the Reynolds Averaging approach. Reynolds Averaged Simulation (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. This is of course an obvious consequence to the fact that taking these higher moments is simply a mathematical endeavor and has no physical contribution what so ever.
Levels of modeling 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 (or time 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” (as explained in the following post: Understanding The k-ω SST Model) because unlike 0-equation and 1-equation models (with exception of 1-equations transport for the eddy viscosity itself as described in a post: Understanding The Spalart-Allmaras Turbulence Model) 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.
The turbulent boundary-layer and the “law of the wall”
RANS is already a “working horse” but has shown poor performance due its inherent limitation 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 hybrid formulations incorporating RANS and LES in certain ways. One of the most popular is Philippe Spalart’s Detached Eddy Simulation (DES) and its variants where RANS is applied to some portion of the boundary layer and large eddies are resolved away from these regions, in free shear flows of which the large eddies are at the order of magnitude as the shear layer and LES may provide extremely reliable information in a high but fair computational cost.
SST-URANS Vs. LES – – Circular cylinder in a cross flow at Re=3.6⋅106
( Iso-surface of Q=S2-Ω2, coloured according to the eddy viscosity ratio)
But DES is essentially LES conducted away from attached flow regions and moreover has many subtleties which are often overlooked and may shift what seems like nice unsteady features to a very erroneous result both qualitatively and quantitatively.
Scale Adptive Simulation (SAS)
In a variety of rectilinear steady flows (ranging from zero-adverse pressure gradient boundary-layers, channel flows, etc’…) RANS models perform well to predict the mean flow statistics and is relatively inexpensive, but the flow physics it can predict to an acceptable physics fidelity is very limited due to the basic fact that most are essentially one-point closures.
An interesting methodology to simulate 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).
Menter-Egorov URANS – “Scale-Adaptive Simulation” (SAS) is based on Rotta’s exact transport equation for kL (from 1970), uses a relation between the integral length scale:
and the diagonal two-point correlation tensor measured at a location x with two probes at distance:
Rotta’s exact transport equation for Ψ=KL reads:
Here the mean velocity is aligned with x-axis and the mean shear with y-axis.
Now enters the most important (and fun?… 🙂 ) part subsequently following the mathematical endeavor in each and every construction of closure transport equations, the surgical identification and simplification by physical reasoning of the terms in the initial transport equation. As the left hand side of the above equation is the advection of Ψ=KL they are identified, and while examining the equation, it is found that what distinguishes the model from other 2-equation eddy viscosity closures is a production term in the second line, namely:
This is actually the mean flow gradient measured at the location of the second probe. Expanding it to a Taylor series:
Rotta postulated the second derivative as negligible and left the third derivative in the expansion. The reasoning behind doing so relies on the observation that in homogenous turbulence the correlation function inside the integral is symmetric with respect to the distance between the fixed and traversing probe (ry). The product of correlation function inside the integral and distance between the fixed and traversing probe (ry) is therefore asymmetric and the integral becomes zero.
Leaving the third derivative as the length-scale determining term was found by Menter and Egorov to be somewhat problematic. First, there is no actual physical reasoning to support such a large contribution from the third derivative.
The understanding that besides the physical objective presented above, leaving just the first derivative does not distinguish the transport equation from any other 2-equation closure methodology leads to the second reason and the cause for Menter-Egorov model variation to include a production term (non-existent in any 2-eq RANS) that reproduces the turbulent spectrum and retains small-scale (high wave-number) behavior due to its actual dependency on a second derivative of the velocity gradient as opposed to rotta’s suggestion to retain third order derivatives which was found inconsistent as it does not retain the “law of the wall”.
The argument in retaining the second derivative of the expansion is that homogenous turbulence can only exist when there is a constant shear (or none), only then (by definition), is the second derivative zero. So the argument is that it is an inhomogeneous term by nature and hence left as a leading order contributor, also found to be consistent with the “law of the wall”.
The KSKL Model
The acknowledgement that still, the integral multiplying the second derivative must be zero under homogeneous flow conditions led Menter and Egorov to assume the ratio of the turbulent length scale to the von Karman length scale as the measure for non-homogeneity:
so the ratio goes to zero for homogeneous flows, and the Taylor expansion after the surgical identification and simplification gets the form:
The equation for Ψ=KL is exactly as rotta’s except for the specific alteration:
The final form as presented by Menter and Egorov is for Φ, the square root of KL, which is proportional to the eddy viscosity, hence the turbulence kinetic energy and the square root of KL transport equations could be transformed to an eddy viscosity transport equation under a straightforward procedure (as explained in: Understanding The Spalart-Allmaras Turbulence Model) and so the 2-equation turbulence model reads:
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.
As such the model is a 2nd generation URANS based 2-eq model (i.e. not dependent explicitly on the step size of the computational grid as in LES – closure is achieved) able to get very good quantitative results for many unsteady flows even on a relatively coarse mesh.
SAS as a scale resolving simulation alternative and incorporation of SAS in hybrid RANS-LES methodology
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 writen 😉 ) 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 model 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 model will essentially perform as RANS on coarser grids.
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)