Philosophies and Fallacies in Turbulence Modeling P. Spalart Turbulence in Engineering Applications...

Philosophies and Fallaciesin

Turbulence Modeling

P. Spalart

Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

H. Lomax M. Strelets

• Paper with same title– To be submitted to Progress in Aerospace Sciences– Soon…

• This talk:– Has the same structure– Covers only a subset of the Fallacies

• (but lists them all)

Companion ArticleTurbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

• Fundamental paradox of turbulence modeling– What does a Reynolds stress mean?– Do/should models have local formulations?

• Philosophies of modeling– Systematic philosophy– Openly empirical philosophy

• Fallacies of modeling– Hard fallacies– Intermediate Fallacies– Soft Fallacies

• Underlying assumptions in turbulence modeling

OutlineTurbulence in Engineering Applications – Nov. 17-21, 2014, UCLA


Average

Fundamental Paradox

• Reynolds (time) averaging defines Reynolds stresses:

• The mean velocity Ui and stress <uiuj> “exist” locally at (x,y,z)

Vorticity and mean streamlines

• The lift signal has considerable modulations• Phase averaging cannot be justified

Character of Vortex Shedding by Cylinder


• Some systems have very small components

Motivation for Fully Time-Averaged Approach


• A boundary layer at high Reynolds number has a very large number of similar eddies

• Is Reynolds averaging now “natural?” Should RANS work well?

Flows with “not as Obviously Disparate” Eddies


DNS of the Ekman layerby R. Johnstone, U. of Southampton

• Classic non-local model: the algebraic model– Outer model in boundary layer nt = 0.02 Ue d* f(y/d)

– Inner model mixing lengthVan Dreist l = k y ( 1 - exp(-yut/[26 ]n ) )

• Modern RANS models avoid ut, and even more Ue, d* and d• Two reasons to prefer a local model:

– Convenience in a CFD code– Physics (see below)

• There are intermediate levels of locality:– Use of the wall distance d, or wall-normal vector n– Both are pre-calculated. n is discontinuous– Both should make the term “dormant” in free shear flows

• In view of Fundamental Paradox, the physics of the locality preference are debatable– Even local models are tested only in large mature regions of turbulence– Sub-regions are coupled by history, transport and diffusion terms

• In incompressible flows, pressure is a “non-local” quantity

Local Formulations for Turbulence ModelsTurbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

Local and Non-Local Quantities


d, or y

Fieldpoint

d

n

Wall

• Exact equation for evolution of Reynolds stress:

– Again all these terms “exist”

• Red terms, production and viscous diffusion, are exact• Other terms are “higher moments” and need modeling

– It is the Closure Problem– The objective is to model each term well, separately– The ordering is NOT an expansion in terms of small or large parameter

• This approach rests on the “Principle of Receding Influence”– Expression coined by Hanjalic & Launder– But there is no reason the higher moments will be easier to model

• The budgets tend to contain several opposing large terms

Systematic Philosophy of RANS Modeling


• Classic: Boussinesq approximation

– Formula has merit, but is not exact in ANY known non-trivial flow

• k and nt are complex functions of the flow field– i.e., not purely local in nature– The cm equation in k-e models is highly empirical

• Other classic to provide nt in algebraic models:– mixing length l = k y (1 - exp(-yut/[26 ]n ) )– The wall distance also used in common transport models

• Terms often come “from thin air,” e.g. cb2 in SA and a1 in SST• More daring:

– Use of time derivative DSij/Dt (Olsen lag model and SARC model)

– Quadratic term [ WikSkj+WjkSki ] (Wilcox-Rubesin, QCR)

Openly Empirical Philosophy


• In principle, the Systematic Philosophy drives strict disciplines– No compensation of errors between terms– Local formulation; no wall distance– Preference against viscous damping functions– Only first derivatives in space and time

• In practice, some disciplines are ignored:– Widespread cancellation between terms

• e.g. anisotropy of pressure-strain and dissipation tensors

– Even some key terms are Openly Empirical• e.g., diffusion terms, especially Daly-Harlow

– Some Reynolds-Stress models use wall distance and normal vector• And many viscous damping functions

• Law of the wall does not apply to stresses, but models expect it!• What is “the best of both worlds?”

– More exact terms, and more successful empiricism!– Model complexity can run away from us, for coding, AND calibration

Boundaries and Bridges Between the PhilosophiesTurbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

• “Isotropy of the diagonal Reynolds stresses”– Isotropy of linear eddy-viscosity model

• “The velocity is a valid input in a model”– Acceleration or pressure gradient are valid inputs in a

model• “Unsteady flows are more difficult than steady

ones”• “Wall functions allow a radical reduction in the

number of grid points”• “The swept-wing Independence Principle applies

to turbulent flow”

Hard Turbulence Fallacies


• This is a common complaint about Boussinesq models– Also called Linear Eddy Viscosity Models, LEVM

• Consider a simple shear flow with U(y)– Write the LEVM stress tensor in axes oriented at an angle q to x:

– The diagonal stresses depend on q!• The statement “the diagonal stresses are isotropic” is meaningless

– Yet, it is found in numerous papers and (good) textbooks• Similarly, calling a LEVM “isotropic” is misleading

– The stress tensor is not isotropic (unless dU/dy = 0)– The anisotropy is merely too simple– The model has two quantities to produce six stresses

“Isotropy of the Diagonal Reynolds Stresses”Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

• This point overlaps with a joint JFM paper with Speziale• It is easy to agree the velocity is not a valid input

– Velocity is not a Galilean Invariant• Model depends on reference frame. Train, or train station?

– But manuscripts appear now and then with it!• Acceleration is invariant between inertial frames…

– However, acceleration does not influence vorticity– “A water-tunnel experiment does not need to stop because of an

earth-quake” (neither does a CFD run!)• An incompressible turbulent flow is insensitive to acceleration

– (with hard boundary conditions)• Therefore, it is very wise to exclude acceleration from any

turbulence model

“The Velocity/Acceleration are Valid Inputs”


• Papers focus on “unsteady” flows, as being more instructive– E.g., airfoil dynamic stall, or channel with pulsed mass flow

• All turbulent flows are unsteady, by nature• Are some flows “more unsteady than others?”

– Because boundary conditions are time-dependent?– Remember the cylinder flow!

• The property of being steady is not Galilean-invariant• Consider turbulence encountering a (“steady”) shock-

boundary layer interaction– Is it exposed to a mild stimulation? – Is it easy to predict?

“Unsteady Flows are Harder than Steady Flows”


• “The Karman constant may depend on flow type and pressure gradient”• “Realizability is an essential quality for a model, and ``weak realizability''

has meaning”• “There exists a well-defined concept of an ``equilibrium'' turbulent flow,

which reveals a relatively simple physical situation”• `` `Artificial’ turbulent flows are relevant test cases”• “It is important for the eddy viscosity to be O(y3) at the wall”• “Obtaining the correct values k and e (or w) is the key to success in a two-

equation model”• “The flat plate boundary layer, unlike the pipe or channel, has constant

total shear stress”• “The two-layer model of wall-bounded flow is a rigorous Matched

Asymptotic Expansion”• “One-equation turbulence models `cannot be complete’ '‘• “Extra strains, such as dV/dx for streamline curvature, are correct empirical

measures to use in a model”

Intermediate Turbulence Fallacies


• True realizability: Reynolds stress tensor is positive-definite– This is of course true for the exact tensor– It is not guaranteed with two-equation models

• The Realizable k-e model has a high status

– It is usually not satisfied by one-equation models– It is not difficult to remedy,

• by adding a multiple of the identity matrix• However, the effect is weak especially at low Mach number

– There is a danger of expecting too much from it• “Weak” realizability: diagonal components are positive

– Consider

– The eigenvalues of this matrix are -1, 1, and 3– This concept depends on the axes used; it is a hard fallacy

“Realizability is an Essential Quality”Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

• The word implies a flow that is “easier to predict”• It has had at least two specific meanings:

– The pressure gradient on a boundary layer is sustained, as expressed by a constant b = d* (dp/dx) / twall

• The choice of word is unhelpful. How about “self-similar?”• These flows still have significant evolution of the turbulence driven by

intense effects (strain, diffusion, pressure term…)• This class of flows is still a valid training ground

– “Production = Dissipation”• P = e in log layer, but not in many “well understood” flows• Many models have corrections that are functions of P / e • In what sense is P / e fundamental? • Much hinges on transfers from one Reynolds stress to another, which

do not affect the TKE k

“Equilibrium Turbulent Flows”


• That nt / n = A y+3 + O(y+4) is an exact result. However,– By definition, this is a viscous region. nt is not separated from n– They enter the momentum equation “on a linear scale”– O(y3), O(y2) or O(y4) behavior is a minor detail

• Some models (both RANS and SGS) are constrained to give O(y3),– But the developments never determine the coefficient A in front of y+3!

“It is Important for the Eddy Viscosity to be O(y3) at the Wall”


Figure:A. Garbaruk

• In the 1970’s and 80’s it was accepted that:– The minimal “description” of turbulence had a velocity scale and a second scale

(length or time)– One-equation models would always need a component similar to algebraic models

• In the 70’s, Secundov in Moscow had a complete model, now nt -92• In 1990, Baldwin & Barth proposed a complete model

– Although it has a serious difficulty at the edge of the turbulent region• In 1992, the Spalart-Allmaras model appeared

– The wall distance is a key input into it,– but not ut, d*, or other typical “algebraic” quantities– The wall distance is a little inconvenient for coding– Not having an internal time scale is a little inconvenient in modeling

• Two-equation modelers take many liberties:– k may not be the true TKE; production may be by vorticity, etc.

• The Boussinesq approximation and nt = cm k2 / e are major assumptions• The choice between e, w and l for second variable is “a matter of taste”

“One-Equation Models will Never be Complete”


• “Homogeneous Isotropic Turbulence is the starting point of RANS modeling”

• “Rapid-Distortion Theory provides valuable, discriminating constraints”

• “The Lumley Invariants contain all the information needed on the anisotropy of the Reynolds-stress tensor”

• “Algebraic Reynolds-Stress Models (ARSM) inherit accuracy from the RST models they are derived from, rigorously”

• “The wall distance and wall-normal vector, and viscous damping functions are serious flaws in a RANS model”

• “The Daly-Harlow Generalized Gradient Diffusion Hypothesis is fully understood”

• “The two-component limit is a valuable, discriminating constraint”

Soft Turbulence Fallacies


• Isotropic turbulence is priceless to study nature of turbulence– Chaos, energy cascade, dissipation, role of viscosity

• It has been the first step of model calibration– TKE decay traditionally obeys a power law: k = A t-p with p ~ 1.2– This sets a basic constant in two-equation models: Ce2 = ( p + 1 ) / p

• The decay power depends on the spectrum for low k– This is the durable part of the spectrum,– By dimensional analysis, p = 2 – 4 / ( 3 + q )– q = 4 gives p = 10 / 7

• But q is arbitrary!– The k4 spectrum is a favorite, but k2 is also respected (Saffman)– Therefore, Ce2 is set based on an arbitrary initial condition– The energy-containing eddies of Isotropic Turbulence are not “natural”

• For different reasons in experiments and in DNS

• This agrees with ideas of Skrbek & Stalp, 2000

“Isotropic Turbulence is the Starting Point of Modeling”Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA

• By M. Dodd and A. Ferrante, U. of Washington• 5123grid, initial Rl = 40, k4 low end of spectrum,• which implies t-10/7 decay

Spectra and TKE Decay in DNS of Isotropic Turbulence


• “Natural” power of k for E(k) appears not to be 4, or 2• Energy is well to the left of where is was injected

Spectra in Experiment of Isotropic Turbulence


Figure: A.Garbaruk

Grid size Structure size?

Van Dyke’s book

• The origin is a short paper of Rodi, 1976– He has not used it recently– The conjecture is that the source terms in the transport equation for the

anisotropy aij are zero:

– Under the source terms, all the Reynolds stresses grow at the same rate– Then, a non-Boussinesq model, giving aij, is linked to a stress-transport model,

through non-trivial “reverse engineering”– In later years, large amounts of algebra were applied

• The problems are:– The purpose of a non-Boussinesq model is to better capture anisotropy in non-

trivial deformations, when more than one stress matters,• but the calibration is done when the anisotropy is not evolving

– We know of no experimental or DNS support for the conjecture• That could have taken the form Daij/Dt<<(Dk/Dt)/k, or << (De/Dt)/e

– Models have progressed since 1976, but this assumption is frozen in time

“Algebraic Reynolds-Stress Models Inherit Accuracy from Differential Reynolds Stress Models, Rigorously”


• Reasons these quantities are “undesirable:”• 1. Convenience and stability

– d is a little difficult to calculate• People have cut corners in codes• For grid blocks, and oblique grid lines, and limiters• Searching is more difficult on massively-parallel machines

– Its derivative is discontinuous– n is discontinuous and hard to calculate– Smooth definitions of “effective distance” from a PDE exist

• Work or Fares and Tucker, and others• n can then be defined as grad(d)• These definitions alleviate the “wire problem” (next slide)• They could be much more efficient on parallel machines

• 2. Physics

“Wall Distance and Wall-Normal Vector are Serious Flaws”


• 2. Physics– Quantities are absent from Reynolds-Stress Equation– Small bodies, such as wires, have excessive impact– However, any empirical attitude recognizes that the physical

influence of a wall is major• Budget of <u’v’> in BL is dominated by pressure-strain

– i.e., by a “wall-reflection,” “non-local” effect

• Empirial model equations are created so wall influence fades– Typical terms are proportional to 1/d2

• In other models, the wall influences the turbulence through the boundary conditions and the diffusion terms

– Is that a natural vehicle for the wall blockage (“splatting”) effect?

– Proposals to eliminate d from one-equation models• They fall back on using the velocity, which is not invariant

– Use of d and n in “legitimate” RST models

“Wall Distance and Wall-Normal Vector are Serious Flaws”


• RANS will outlive us, and is a highly justified field of work– In transportation, pure LES will not be possible for decades, if ever (DNS?)– Within hybrids, the switch RANS-> LES will occur earlier, in the attached BL

• This may lead to RANS models aimed at only boundary layers

• The beauty of RANS research can be… hidden!– It IS there, and so is discipline (invariance, well-posedness, sensitivity)– The rewards for successful modeling work are more than adequate

• Steps based on analytical “Turbulence Facts” are attractive…– But it is possible (easy?) to be seduced by them

• DNS has not had the impact on RANS we all hoped for• Valid question: does an excellent RANS turbulence model exist?

– (with any number of equations)– Or is there a “glass ceiling” to accuracy?– The answer inspires the choice of calibration cases

• If “yes,” the cases can be invoked in any order• If “no,” identify a “cloud” of meaningful cases and ignore “corners of the envelope”

• Also valid: is a respectable model understood to be universal?– Or can it be restricted to a class of flows? (for instance, boundary layers)

Underlying Assumptions in Turbulence Research


Detached-Eddy Simulation


LESRANS

Philosophies and Fallacies in Turbulence Modeling P. Spalart Turbulence in Engineering Applications...

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Transcript of Philosophies and Fallacies in Turbulence Modeling P. Spalart Turbulence in Engineering Applications...