Philosophies and Fallacies in Turbulence Modeling

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Philosophies and FallaciesinTurbulence Modeling
Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
P. Spalart
H. Lomax
M. Strelets
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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)

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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

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

• Reynolds averaging defines Reynolds stresses
• The mean Ui and stress ltuiujgt exist locally

Real flow
Reynolds averaged flow
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Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
Fundamental Paradox

• Reynolds (time) averaging defines Reynolds
  stresses
• The mean velocity Ui and stress ltuiujgt exist
  locally at (x,y,z)
• Vorticity and mean streamlines

Average
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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

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Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
Motivation for Fully Time-Averaged Approach

• Some systems have very small components

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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
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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

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Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
Local and Non-Local Quantities
Field point
d

• d, or y

n
Wall
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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

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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

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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 WikSkjWjkSki (Wilcox-Rubesin,
  QCR)

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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

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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

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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

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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

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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

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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)
  shock-boundary layer interaction
• Is it exposed to a mild stimulation?
• Is it easy to predict?

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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

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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

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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

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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 y3 O(y4) 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 y3!

Figure A. Garbaruk
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Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
One-Equation Models will Never be Complete

• In the 1970s and 80s 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 70s, 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

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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

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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

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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

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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

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Grid size
Structure size?
Van Dykes book
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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/Dtltlt(Dk/Dt)/k,
  or ltlt (De/Dt)/e
• Models have progressed since 1976, but this
  assumption is frozen in time

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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

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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 ltuvgt 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

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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-gt 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)

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Turbulence in Engineering Applications Nov.
17-21, 2014, UCLA
Detached-Eddy Simulation
RANS
LES