It’s quite amazing how 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 by many predicted to keep doing so for the next 20 years if not more.
Moore’s Law applied to CFD
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.
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.
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.
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.
The turbulent boundary-layer and the “law of the wall”
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
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).
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)
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 original PANS model is 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.
Direct Numerical Simulation (DNS)
My first actual encounter with DNS was while researching for my thesis relating to the role of hairpins in transition and turbulence(specifically originating from bypass transition mechanism). ChannelFlow code as simple as it was made me feel ever so powerful in my direct confrontation with turbulence… 😉
Turbulence phenomena is very precisely described by a seemingly simple set of equations, 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.
Saying all that, it is not expected that DNS will take on vital role in the engineering design process, where many designs are to be evaluated working through a repetitive cycle of obtaining a CAD geometry–> grid generation–>Solving the equation–>post-processing the results–>optimization decisions.
Nonetheless, DNS shall find its place in the industrial CFD community for specialized research as it does in the academy, where on the line of an academic study which lasts up to approximately 5 years only a few high-fidelity simulations are conducted.
Large-Eddy Simulation (LES) and hybrid RANS-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…
Modeling of LES is formally described by the application of spatially filtering NSE. An explicit approach would explicitly apply a filter with some kind of shape (may it be cutoff, top hat, etc…). subsequently, a model is devised to capture the effect of under-resolved length-scales. The most common representation, is a linear stress-strain relation relying on the Boussinesq hypothesis and the eddy viscosity concept. The first and possibly still the most popular is the Smagorinsky model. Applying the Smagorinsky model to flows other than those it was tuned for, shall prove out of its range of applicability consequence of its many shortcomings, fully explained in my former post That’s a Big W(H)ALE as well as the remedies to overcome these shortcomings from a purely physical perspective.
Models such as these are termed “explicit SGS Models” as the filter and its shape are “clearly” defined (Its effect not quite though…). Other popular explicit modelling procedures include:
- Dynamic models (Going Dynamic I )
- Scale similar and mixed models (Bardina et al.)
- Structure function models (Lesieur – great book by the way, very recommended)
- Deconvolution methods (Stoltz et al.)
Another route for modelling the effect of unresolved scales is found through the utilization of higher order numerical schemes to take the role of the explicit filter in the aim of adding dissipation only in the high wave number range (small and unresolved scales) – termed Implicit LES (ILES). The first of such method was MILES (F. Grinshtein, also followed by a good book on the subject of ILES).
Hybrid RANS-LES and its engineering added value
As LES shall remain too expensive in the following few decades for the ever increasing number of engineering complexities (e.g. complete aircraft wing), researchers have shifted much of the attention and effort to hybrid formulations incorporating RANS and LES in certain ways. Due to the fact that computational cost of LES is practically independent on the Reynolds number for free shear ﬂows, only weakly dependent on the Reynolds number for the outer portion of the turbulent boundary layer, but becomes strongly dependent on the Reynolds number for the innermost layer (the viscous sublayer, the buffer layer and the initial part of the log layer), in most hybrid RANS-LES methods RANS is applied for an inner portion of the boundary layer and large eddies are resolved away from these regions by an LES (e.g. WMLES).
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 incorporation of SRS in engineering process
In order for SRS to be best incorporated in engineering design process there are some challenges to overcome, most of which are related to LES rather than second generation URANS, based on RANS methodology which is 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.
Optimization and sensitivity analysis
Engineering design process is based on an iterative design achieving the best product through assessing a current design by optimization methodologies such as local sensitivity analysis, by which gradients of design parameters are calculated subsequently to be employed in gradient-based optimization algorithms.
In order to being able to use LES in such quantifications of design parameters it needs to be incorporated with tools of sensitivity analysis to measure how uncertainty factors affect the performance of the design.
The problem is that LES is a non-linear dynamical system, hence suffers from chaotic behavior. Local calculations of quantities of interest of which initial conditions slightly depart, exponentially diverge as time advances. A robust methodology to avoid uncertainty calculations divergence is mandatory if LES is to participate in the engineering design process.
The “strange attractor” – chaos and fractals (Lorenz)
Geometry, grid generation and numerical schemes
In order for LES to come forth on its future vital role, many adjustments and advancements to current dominating LES approaches should be conducted. In essence, what differs practical engineering applications from their academic counterparts is the level of geometry complexity. Unstructured meshing for complex geometries has been dominating industrial CFD and from an LES standpoint this means that large errors due to commutation of non-commutative operations may hamper results accuracy substantially.
Advancements of Immersed Boundary Method (IBM), in which the boundaries of the body do not conform to the grid, the governing equations are discretized on fixed meshes and applying boundary conditions requires modifying the equations in the vicinity of the solid boundary by means of a forcing function that reproduces the effect of the boundary, are promising as far of high fidelity simulations of complex geometry and especially for moving meshes are concerned.
A snapshot of Large Eddy Simulation of a 5-bladed
rotor wake in hover with a novel multiblock IBM
(by Technion CFD Lab – S. Frankel)
Some new advancements in mixed-models (such dynamic and Smagorinsky) based on the integral formulation of the LES equation (F. M. Denaro) alleviate some of commutation problematic issues and allow for a much more accurate filtering.
Moreover, of true importance is the increasing the level of automation. As HPC shall keep obeying Moore’s law in its advancement, CFD workflows shall suffer tremendously from the “human-in-the-loop” syndrome, where the practitioner is too much involved, especially in the geometry accommodation and grid generation phases of the design and analysis.
Adaptive grid-generation for SRS are also a challenge. While in RANS grid adaptation is aimed only on reducing numerical error, for LES it is intended also to improve SGS model errors and increase the fraction of resolved motions. Suggestions to alleviate the difficulty are strongly related to the fact that standard algebraic eddy-viscosity modeling approach render LES as unclosed in the sense that the filter to be applied is not grid-independent. One exciting route of such by SB Pope suggesting adaptation aiming on resolving a user-deﬁned fraction of the kinetic energy, and also presented an incorporation of such in dynamic modeling.
High Power Computing (HPC)
The effectiveness and impact of CFD on the engineering design process is extremely dependent on the power and availability of modern HPC systems. During the last decades, CFD codes were formulated using message passing (MPI) software models which match nowadays parallelism efficiently. As future route and prevailing computing hardware, memory architecture (hierarchical not supported by MPI) and network connecting is not a-priori known new algorithms have to be supportive and advance hand to hand with computing resources.
Numerical schemes such must also support tremendous parallelism in future exascale computing. Schemes involving global operations shall not prevail do to obvious bottlenecking.
Worldwide top HPC and its utilization
The second issue relates to the fact that in order to utilize such computational advancements, methodologies for SRS should be developed as to be also used outside the academy. As much as it is important that novel modelling techniques shall be validated and tested on simple canonical problems (e.g. ZPGBL/couette/channel/pipe flows) which lend themselves to detailed assessment, they should be developed to be also applied to real engineering problems.
It is no coincidence that the k-ω SST 2-equation turbulence model (F. Menter) Detached-Eddy Simulation (DES) (P. Spalart) and WALE LES model (F. Nicoud and F. Ducros) have gained such popularity. It is the fact that each of them was devised intentionally to perform well for industrial applications, that made them such. A good example is non-local operations which find their way to many LES formulations. Their use in commercial CFD code environment is near to impossible.
Democratization of SRS
I read quite an interesting post (by Keith Hanna – Mentor Graphics) about the “Democratization of CFD“. Referring to SRS, it is quite obvious that in order for LES to be widespread in the design process, it clearly needs to be much more accessible to non-proficient practitioners. In my post “Let’s LES” I have reviewed some of LES mandatory set of tools without which the credibility of the simulation is doubtful at best. When a non-proficient practitioner tries to perform an LES, there are many instances of which being able to construct an animation of a time-varying flow that looks like a turbulent flow seems very satisfying. However, this offers no guarantee that the appropriate grid resolution has been used, spatial and temporal schemes have been selected, boundary conditions (especially time-varying, turbulent containing inflow conditions) are proper, etc’… CFD practitioners have to be educated to control a much different set of tools than those they were used to with RANS (and other low fidelity methodologies) to actually achieve the added benefit that LES could provide.
So to conclude…
even though it’s safe to predict that SRS (and especially LES) shall not replace RANS in the near future, the level of physical fidelity achieved by SRS shall have a growing impact on engineering design process.
Returning to Moore’s law prediction it could be assumed that LES is going to take more and more of a vital role in engineering design process, being ever so attractive as its level of fidelity is such that it combines the advantages of simulations along with reliability features of experiments. This allows the engineer to build up his confidence while extracting high fidelity realizable results, such that the margin of safety could be tightening for the few more percentages of optimality which are the hardest to achieve.
For this forecast to become reality, the one conclusion shared by most authors was that sincere confrontation with SRS challenges should be conducted while also taking under consideration its practicality to engineering design process.