Turbulence models for CFD in the 21st century
by
Brian Spalding, of CHAM Ltd
October, 2000
Invited lecture presented at ACFD 2000, Beijing
Click here for start of lecture
Abstract
The two approaches to turbulence modelling
Both Osborne Reynolds (1884) and Ludwig Prandtl (1925) regarded turbulence
as an expression of
the nearrandom intermingling of sizeable fragments of unlike fluid,
which, during a succession of brief encounters, tended to equilibrium.
However, their concept found no place in the family of turbulence
models springing from Kolmogorov's (1942) proposal to attend only
to statistical measures of the turbulent motion, such as
energy and frequency.
The interminglingfragments idea was nevertheless preserved in the
models of Spalding (1971) and Magnussen (1976)
("eddybreakup" and "eddydissipation", respectively)
which are still used for combustion simulation.
It also featured in Spalding's
(1987) "twofluid" model of turbulence; and it is essential to the
"multifluid" models of turbulence (Spalding, 1996) which are the
subject of the present lecture.
Why the Kolmogorov approach has been popular until now
Limitations of computing power, and the seeming simplicity of the
associated (1877) effectiveviscosity concept of Boussinesq (1877),
favoured adoption of the Kolmogorov rather
than the ReynoldsPrandtl approach; and this road has now become so
welltrodden
that most CFD practitioners suppose, wrongly, that it is the only
one which is open.
This would not matter if Kolmogorovtype models, for example
kepsilon
(Harlow and Nakayama (1968)), allowed computation
of the "probability density functions" needed for the
simulation of nonlinear processes such as radiation and chemical
reaction; or if they could comprehend such real processes as
"unmixing"; but they do not.
Why the ReynoldsPrandtl approach is likely to be favoured from now on
"Interminglingfragments" models of the kind conceived by Reynolds and
Prandtl, do however permit these things; and the computing
power needed for using them is easily available nowadays.
The lecture will explain how such "multifluid models" may be:
 formulated
 calibrated
 utilised for simulating engineering processes and equipment,
 subjected to numericalaccuracy tests, and
 further developed.
Similarities to, and differences from, the "pdftransport" models of
Dopazo and O'Brien (1974), and of Pope (1982), will be pointed out.
Contents
Click here for a historical overview
 Alternative concepts of turbulence
 Boussinesq's enlargedviscosity ("thicksoup") concept, and
 Reynolds interminglingfragments ("stew") concept.
 An enlargedviscosity model: LVEL
 Where enlargedviscosity models fail
 Why interminglingfragments models (IFMs, MFMs) can do better
 How multifluid models (MFMs) work
 Calibrating MFMs
 Extending MFMs
 Distinguishing MFMs from other models
 Conclusions
 References
1. Alternative concepts of turbulence
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2. An enlargedviscosity model: LVEL
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3. Where enlargedviscosity models fail
 There are however many practicallyimportant turbulentflow
processes which defy simulation by any BoussinesqKolmogorovtype
(ie "thicksoup")
model. One of these will now be described.
 A saltwater layer, lying below fresh water, is heated for a short
time by an electric current.
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___________ ___________ ___________
  * * ** * *   
 fresh   * * * * *  
  * * ** * *  
***********  * * * **  ***********
** salty ** * * ** * *  ***********
***********  * * * * * ***********
  
start later later still
unmixed mixed ? unmixed again !
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 Mixing appears to take place soon after heating equalises the
densities.
 But it is macro not micromixing; and, because the Prandtl and Schmidt
numbers differ,the
"intermingling fragments" transfer heat by conduction more
rapidly than they exchange salt by way of diffusion.
 Therefore "unmixing" takes place.
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 This process can be correctly simulated by even the first of the
interminglingfluids models to be expressed by way of diffferential
equations (Spalding, 1987), as will be demonstrated during the
oral presentation.
Example 2: Twofluid simulation of the mixing/unmixing experiment
Only those CFD models which incorporate the Reynolds,
interminglingfragments, concept can simulate such phenomena.
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 Related phenomena which require IFMs for their simulation include:
 the reduction in the shear stress in a turbulent boundary layer
when it passes over a convexly curved wall;
 the "bumpiness" of lowaltitude airplane rides over sunheated
terrain;
 the mixing of blade wakes with betweenblade gases in turbomachines
 the RanqueHilsch effect;
 almost all processes involving chemical reaction in turbulent
gases.
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4. Why interminglingfragment models (IFMs) can simulate a wider
range of phenomena than EVMs
 Intermingingfragments models, or "multifluid models" (MFMs) to
call them by a more common name, compute the "PDFs".
This acronym serves equally well for:
 the "probabilitydensity functions", which feature in
mathematical representations of the fluctuations with time of
instantaneous values of velocity, temperature or other
practicallyinteresting properties; or
 the "populationdistribution frequencies", which is the
term used in the MFM literature.
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 PDFs of the second kind can be regarded as discretised
versions of PDFs of the first kind.
 They take the form of
histograms, whereby:
 the ordinate represents the proportion of the
population
 lying within the corresponding abscissa interval;
 and the abscissa stands for the populationdistinguishing
attribute in question, for example:
 temperature
 fuel concentration
 vertical velocity
 etc
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^ __ a onedimensional "PDF""
 нн __
 нн __ нн
 нн нн__ нн
 нн __ннннн нн
 нн__ нннннннн__ нн
 нннн __ннннннннннн__ нн
 __ннннн__ннннннннннннннннн__нн__
__ннннннннннннннннннннннннннннннннннн____
 populationdefining attribute >
for example temperature or concentration
^
populationmean value _
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Knowledge of PDFs is essential whenever important phenomena depend
in a complex fashion on the populationdistinguishing attribute.
 For example, the thermal radiation or the rate of chemical
reaction will be quite different for:
 a gas mixture having the PDF just shown, and
 a gas which is completely mixed and so possesses the
populationmean temperature or concentration.
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 In the salinelayer example, the vertical velocity of the
fluid fragments is negative for the heavier and positive for the
lighter members of the population.
 Kolmogorovtype models compute only populationmean values (although
they are sometimes supplemented by guesses about the PDFs).
 This is why they simulate combustion processes (for example)
inadequately.
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5. How multifluid models (MFMs) work
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6. Calibrating MFMs
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7. Extending MFMs
 Whereas the research on Kolmogorovtype models has been conducted
extensively, throughout the world, since the late 1960s, that on
multifluid models has scarcely started.
 Opportunities exist therefore for the further development of such
models in several directions, including:
 experimental, devoted to devising situations like that of
the "puffjet" which throw light on the postulated micromixing
mechanisms;
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 conceptual, conceiving new formulae for say:
 the influences of Prandtl and Schmidt numbers on the
micromixing rates;
 the rates of fragmentsize increase as a result of
collision and diminution as a consequence of meanflow
shear;
 the 'scattering' into the y and zdirectionvelocity
intervals which results when fragments having differing
xdirection velocities collide;
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 mathematical, especially in the development of the
creating and handling of unstructured and adaptive "population
grids", which:
 use different numbers of fluids (ie abscissa intervals)
at different positions and times,
 vary these as the calculation proceeds, guided by
optimization rules for:
 economy and
 accuracy;
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 presentational, so enabling the stillunfamiliar MFM
concepts to be grasped by those who can benefit from them, for
example by providing:
 means of displaying the development of the PDF shapes as the
calculation proceeds;
 applications to familiar flow, heattransfer and
chemicalreaction systems which will reveal how
multifluid models agree with conventional models where
the latter are valid, but with experimental data whare the
latter do not;
 tutorials and selfstudy material.
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 Applicationoriented, because, although much research is desirable,
there is no need to
delay use of MFMs in engineering practice until it has been
completed.
The reason is that such tests as have already been conducted have
confirmed the inherent plausibility of the multifluid approach;
whereas to rely on models which neglect the
intermingledfragments aspect of turbulence is inherently unsafe.
The kepsilon and eddybreakup models were adopted by industry on the basis of
much flimsier evidence than now exist for MFM; but someone had to be the first.
Who will be the MFMapplication pioneer?
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8. Distinguishing MFMs from other models
 The division of turbulence models into just two types
cannot of course do justice to the rich variety of turbulence models
which have been invented. It is therefore proper to make a few
further explanatory remarks, as follows.
 Multifluid models have much in common with, but are distinct
from, the "PDFtransport" models deriving from the work of
Dopazo and O'Brien (1974) and Pope (1982). Because the latter employ a Monte Carlo
method of
solution, they appear to lack some conceptual and practical
advantages which the "discretisedPDF" nature of MFM offers.
However, given unlimited computer time and care to employ
precisely the same micromixing formulae, MFM and PDFtransport
should produce the same answers.
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 There exist Kolmogorovtype models which do not employ
the enlargedviscosity idea. These are the Reynoldsstress models
which have been proposed and employed by many workers, for
example Launder. Reece and Rodi (1975).
They still however attempt to deduce all interesting phenomena from
the distributions of statistical quantities, from which
the needed PDFs can not be derived.
No ReynoldsStress model could predict the "unmixing" behaviour
reported in section 3.
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 Finally direct numerical simulation should be mentioned,
not because DNS is a turbulence model but in order to lead to
the following remark:
Whereas DNS has sometimes been used as means of deriving the
constants and functions of Kolmogorovtype models, it could now
equally well be used for testing and augmenting the micromixing
hypotheses of MFM.
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9. Conclusions
The argument presented in the foregoing lecture will now be
summarised, as follows:
 Of the two main approaches to turbulence modelling, namely
 BoussinesqKolmogorov (i.e. "enlargedviscosity") and
 ReynoldsPrandtl (i.e. "interminglingfragments"),
it is the second
which is better able to represent physical reality.
 ReynoldsStress models, although they do not use the Boussinesq
enlargedviscosity notion, are no better able to provide the needed
PDFs.
 PDFtransport models based upon Monte Carlo methods, although
directed at the right target, have builtin limitations which will
continue to prevent their widespread use.
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 Because of:
 the attractiveness of Kolmogorov's guess that statistical quantities
might suffice, coupled with
 the limited computing power available when CFD first started,
Reynolds' interminglingfragments approach to turbulence has been
almost entirely neglected.
 Now, however, computing power is more than adequate; and sufficient
work has been done to demonstrate its practicability and promise.
 The author recommends that approach, as currently embodied in MFM,
as the better basis for CFD in the TwentyFirst Century.
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10. References
contents
 J Boussinesq (1877) "Theorie de l'ecoulement tourbillant";
Mem. Pres. Acad. Sci. Paris, vol 23, 46
 C Dopazo and EE O'Brien (1974) Acta Astronautica vol 1, p1239
 FH Harlow & PI Nakayama (1968) "Transport of turbulenceenergy
decay rate"; Los Alamos Sci Lab U Calif report LA 3854
 AN Kolmogorov (1942) "Equations of motion of an incompressible
turbulent fluid"; Izv Akad Nauk SSSR Ser Phys VI No 12, p56
 BE Launder, GJ Reece, W Rodi (1975) "Progress in the development of a
ReynoldsStress closure" JFM, vol 68, p 537
 BF Magnussen and BH Hjertager (1976) "On mathematical modelling of
turbulent combustion with special emphasis on soot formation
and combustion". 16th Int. Symposium on Combustion, pp 719729
The Combustion Institute
 SB Pope (1982) Combustion Science and Technology vol 28, p131
Springer Verlag, New York, 1980, p115
 L Prandtl (1925) "Bericht ueber Untersuchungen zur ausgebildeten
Turbulenz"; ZAMM vol 3, pp 136139, 1925
 O Reynolds (1874) "On the extent and action of the heating
surface of steam boilers";
Proc. Manchester Lit Phil Soc, vol 8, 1874
 DB Spalding (1971) "Mixing and chemical reaction in confined
turbulent flames";
13th International Symposium on Combustion, pp 649657
The Combustion Institute
 DB Spalding (1987) "A turbulence model for buoyant and combusting
flows"; International J. for Numerical Methods in Engineering
vol 24, pp 123
 DB Spalding (1994) Poster session, International Heat Transfer
Conference, Brighton, England
 DB Spalding (1996) "Multifluid models of Turbulence; Progress and
Prospects; lecture CFD 96, the Fourth Annual
Conference of the CFD Society of Canada, June 2  6, 1996,
Ottawa, Ontario, Canada
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Turbulencemodelling highlights through four halfcenturies
 Nineteenth century, second half
 Boussinesq (1877) proposes the "enlargedviscosity" approach:
"A turbulent fluid is like a thick soup"
 Reynolds (1874) introduces the "interminglingfragments" approach:
"A turbulent fluid is more like a stew"
(They did not actually use those words)
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 Twentieth century, first half
 Prandtl (1925) uses the "interminglingfragments" approach
(for his "mixinglength hypothesis"), but, for want of other
mathematical tools, casts the result in "enlargedviscosity"
terms.
 Kolmogorov (1942) pays no attention to intermingling fragments
at all, but devises the first means for computing the viscosity
enlargement from transport equations for turbulence and
frequency.
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 Twentieth century, second half
 The primitive computers of the late '60s and early '70s enable workers in Los
Alamos, Imperial College and elsewhere to devise software which solves the transport
equations; so CFD is born.
 CFDforcombustion specialists are forced to adopt the
"intermingling" fragments approach in order to fit the facts.
 CFDformultiphaseflow specialists show how the multifluid
transport equations can be solved .
 Computers increase in power; but most turbulence modellers continue
to use powerrestricted concepts.
 Nevertheless the first steps are taken to develop multifluid
models which make use of both the Boussinesq and Reynolds
concepts.
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 Twentyfirst century, first half
 October, 2000: Audiences in Beijing and Shanghai are presented with the
argument that, since:
 more than adequate computing power is now widely available,
 already one generalpurpose CFD code (which can be
accessed via
Internet) has MFM built into it,
 the superior plausibility of MFM for many processes has already been
demonstrated,
it is time at last to switch from Boussinesq to Reynolds.
 Thereafter, ?????
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Links to explanations of the micromixing hypothesis of MFM
contents