Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Philosophies and Fallacies
in
Turbulence Modeling
P. Spalart
H. Lomax
M. Strelets
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Companion Article
• 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)
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Outline
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
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
Average
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Character of Vortex Shedding by Cylinder
• The lift signal has considerable modulations
• Phase averaging cannot be justified
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Motivation for Fully Time-Averaged Approach
• Some systems have very small components
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Flows with “not as Obviously Disparate” Eddies
• A boundary layer at high Reynolds number has a very large
number of similar eddies
• Is Reynolds averaging now “natural?” Should RANS work well?
DNS of the Ekman layer
by R. Johnstone, U. of Southampton
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Local Formulations for Turbulence Models
• 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/[26n]) )
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Local and Non-Local Quantities
Field
point
d
d, or y
n
Wall
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Outline
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Systematic Philosophy of RANS Modeling
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Openly Empirical Philosophy
• 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/[26n]) )
– 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)
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Boundaries and Bridges Between the Philosophies
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Outline
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Hard Turbulence Fallacies
• “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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Isotropy of the Diagonal Reynolds Stresses”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“The Velocity/Acceleration are Valid Inputs”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Unsteady Flows are Harder than Steady Flows”
• 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”) shockboundary layer interaction
– Is it exposed to a mild stimulation?
– Is it easy to predict?
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Intermediate Turbulence Fallacies
• “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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Realizability is an Essential Quality”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Equilibrium Turbulent Flows”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“It is Important for the Eddy Viscosity to be O(y3) at the Wall”
• 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
Figure:
A. Garbaruk
• 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!
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“One-Equation Models will Never be Complete”
• 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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Soft Turbulence Fallacies
• “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”
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Isotropic Turbulence is the Starting Point of Modeling”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Spectra and TKE Decay in DNS of Isotropic Turbulence
• By M. Dodd and A. Ferrante, U. of Washington
• 5123grid, initial Rl = 40, k4 low end of spectrum,
• which implies t-10/7 decay
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Spectra in Experiment of Isotropic Turbulence
Figure: A.
Garbaruk
• “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
Grid size
Structure size?
Van Dyke’s book
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Algebraic Reynolds-Stress Models Inherit Accuracy from
Differential Reynolds Stress Models, Rigorously”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“Wall Distance and Wall-Normal Vector are Serious Flaws”
• 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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
“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
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Underlying Assumptions in Turbulence Research
• 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)
Turbulence in Engineering Applications – Nov. 17-21, 2014, UCLA
Detached-Eddy Simulation
RANS
LES