“predictions are very difficult, especially about the future…” – Richard Feinmann
Turbulence modelling is considered by many as witchcraft, by others as the art of producing physics out of chaos. The above quote by Niels Bohr, generally to remind the impossibility in predicting future events in life, ever so affected by and coupled with our surrounding and origin, incoroporating circular cause and effect reciprocity, acting so unexpected that it seems one may only put his trust in god or the dice… Such is turbulence!
A full description of the phenomena is entangled in a seemingly simple set equation, the Navier-Stokes equations, their nature is such that analytic solutions to even the most simple turbulent flows can not be obtained and resorting to numerical solutions seems like the only hope.
But the resourcefulness of the plea to 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”.
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 reagion inviscid flow calculations.
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 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.
A snapshot of Large Eddy Simulation of a 5-bladed rotor wake in hover with a novel multiblock IBM
(by Technion CFD Lab)
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.
A higher level of RANS modelling is achieved by not invoking the Bousinesq hypothesis and directly constructing differential transport equations for the Reynolds stresses themself. This method is termed Reynolds Stress Modeling (RSM). As much as RSM has a wider range of applicability than lower level RANS closure models, it comes with a sharp increase in computational effort and of numerical robustness.
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.
All of the above leads to the conclusion that an optimal modelling approach involves both physical fidelity of level of flow field complexity and the computational resources at hand.
As such, researchers have shifted much of the attention and effort to hybrid formulations incorporating RANS and LES in certain ways. In some hybrid RANS-LES methods, such as Detached-Eddy Simulation and its variants, RANS is applied for a portion of the boundary layer and large eddies are resolved away from these regions by an LES.
Other hybrid RANS-LES models are termed Zonal hybrid RANS-LES methods, as they solve clearly distinct subdomains of RANS and LES formulation with the main issue found in providing a consistent blending at interface of such subdomains.
Another addition, somewhat more of an infrastructure to modelling of intermediate cost to accuracy consideration is the Partially-Averaged Navier-Stokes (PANS) Method by S. Girimaji et al.
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.
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.
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.