philosophies and fallacies in turbulence modeling p. spalart turbulence in engineering applications...
TRANSCRIPT
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
Turbulence 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• Some systems have very small components
Motivation for Fully Time-Averaged Approach
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
d, or y
Fieldpoint
d
n
Wall
• 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
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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
• 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
• “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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• “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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• “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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• “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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Detached-Eddy Simulation
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
LESRANS