Turbulence models for CFD in the 21st century
Brian Spalding, of CHAM Ltd
Invited lecture presented at ACFD 2000, Beijing
Click here for start of lecture
The two approaches to turbulence modelling
Both Osborne Reynolds (1884) and Ludwig Prandtl (1925) regarded turbulence
as an expression of
the near-random 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 intermingling-fragments idea was nevertheless preserved in the
models of Spalding (1971) and Magnussen (1976)
("eddy-break-up" and "eddy-dissipation", respectively)
which are still used for combustion simulation.
It also featured in Spalding's
(1987) "two-fluid" model of turbulence; and it is essential to the
"multi-fluid" 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) effective-viscosity concept of Boussinesq (1877),
favoured adoption of the Kolmogorov rather
than the Reynolds-Prandtl approach; and this road has now become so
that most CFD practitioners suppose, wrongly, that it is the only
one which is open.
This would not matter if Kolmogorov-type models, for example
(Harlow and Nakayama (1968)), allowed computation
of the "probability density functions" needed for the
simulation of non-linear processes such as radiation and chemical
reaction; or if they could comprehend such real processes as
"un-mixing"; but they do not.
Why the Reynolds-Prandtl approach is likely to be favoured from now on
"Intermingling-fragments" 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 "multi-fluid models" may be:
- utilised for simulating engineering processes and equipment,
- subjected to numerical-accuracy tests, and
- further developed.
Similarities to, and differences from, the "pdf-transport" models of
Dopazo and O'Brien (1974), and of Pope (1982), will be pointed out.
Click here for a historical overview
- Alternative concepts of turbulence
- Boussinesq's enlarged-viscosity ("thick-soup") concept, and
- Reynolds intermingling-fragments ("stew") concept.
- An enlarged-viscosity model: LVEL
- Where enlarged-viscosity models fail
- Why intermingling-fragments models (IFMs, MFMs) can do better
- How multi-fluid models (MFMs) work
- Calibrating MFMs
- Extending MFMs
- Distinguishing MFMs from other models
1. Alternative concepts of turbulence
next3, back or contents
2. An enlarged-viscosity model: LVEL
next7, back or contents
3. Where enlarged-viscosity models fail
- There are however many practically-important turbulent-flow
processes which defy simulation by any Boussinesq-Kolmogorov-type
model. One of these will now be described.
- A salt-water layer, lying below fresh water, is heated for a short
time by an electric current.
next8, back or contents
___________ ___________ ___________
| | |* * ** * * | | |
| fresh | | * * * * *| | |
| | |* * ** * *| | |
|***********| | * * * ** | |***********|
|** salty **| |* * ** * * | |***********|
|***********| | * * * * *| |***********|
----------- ----------- -----------
start later later still
un-mixed mixed ? un-mixed again !
next9, back or contents
- Mixing appears to take place soon after heating equalises the
- But it is macro- not micro-mixing; and, because the Prandtl and Schmidt
"intermingling fragments" transfer heat by conduction more
rapidly than they exchange salt by way of diffusion.
- Therefore "unmixing" takes place.
next10, back or contents
- This process can be correctly simulated by even the first of the
intermingling-fluids models to be expressed by way of diffferential
equations (Spalding, 1987), as will be demonstrated during the
Example 2: Two-fluid simulation of the mixing/unmixing experiment
Only those CFD models which incorporate the Reynolds,
intermingling-fragments, concept can simulate such phenomena.
next11, back or contents
- 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 low-altitude airplane rides over sun-heated
- the mixing of blade wakes with between-blade gases in turbo-machines
- the Ranque-Hilsch effect;
- almost all processes involving chemical reaction in turbulent
next12, back or contents
4. Why intermingling-fragment models (IFMs) can simulate a wider
range of phenomena than EVMs
- Interminging-fragments models, or "multi-fluid models" (MFMs) to
call them by a more common name, compute the "PDFs".
This acronym serves equally well for:-
- the "probability-density functions", which feature in
mathematical representations of the fluctuations with time of
instantaneous values of velocity, temperature or other
practically-interesting properties; or
- the "population-distribution frequencies", which is the
term used in the MFM literature.
next13, back or contents
- PDFs of the second kind can be regarded as discretised
versions of PDFs of the first kind.
- They take the form of
- the ordinate represents the proportion of the
- lying within the corresponding abscissa interval;
- and the abscissa stands for the population-distinguishing
attribute in question, for example:
- fuel concentration
- vertical velocity
next14, back or contents
^ __ a one-dimensional "PDF""
| |нн| __
| |нн| __ |нн|
| |нн| |нн|__ |нн|
| |нн| __|ннннн| |нн|
| |нн|__ |нннннннн|__ |нн|
| |нн|нн| __|ннннннннннн|__ |нн|
--- population-defining attribute ---->
for example temperature or concentration
population-mean value _|
Knowledge of PDFs is essential whenever important phenomena depend
in a complex fashion on the population-distinguishing 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
population-mean temperature or concentration.
- In the saline-layer example, the vertical velocity of the
fluid fragments is negative for the heavier and positive for the
lighter members of the population.
- Kolmogorov-type models compute only population-mean values (although
they are sometimes supplemented by guesses about the PDFs).
- This is why they simulate combustion processes (for example)
5. How multi-fluid models (MFMs) work
6. Calibrating MFMs
7. Extending MFMs
- Whereas the research on Kolmogorov-type models has been conducted
extensively, throughout the world, since the late 1960s, that on
multi-fluid 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 "puff-jet" which throw light on the postulated micro-mixing
- conceptual, conceiving new formulae for say:
- the influences of Prandtl and Schmidt numbers on the
- the rates of fragment-size increase as a result of
collision and diminution as a consequence of mean-flow
- the 'scattering' into the y- and z-direction-velocity
intervals which results when fragments having differing
x-direction velocities collide;
- mathematical, especially in the development of the
creating and handling of un-structured and adaptive "population
- 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
- presentational, so enabling the still-unfamiliar 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
- applications to familiar flow, heat-transfer and
chemical-reaction systems which will reveal how
multi-fluid models agree with conventional models where
the latter are valid, but with experimental data whare the
latter do not;
- tutorials and self-study material.
- Application-oriented, because, although much research is desirable,
there is no need to
delay use of MFMs in engineering practice until it has been
The reason is that such tests as have already been conducted have
confirmed the inherent plausibility of the multi-fluid approach;
whereas to rely on models which neglect the
intermingled-fragments aspect of turbulence is inherently unsafe.
The k-epsilon and eddy-break-up 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 MFM-application pioneer?
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.
- Multi-fluid models have much in common with, but are distinct
from, the "PDF-transport" models deriving from the work of
Dopazo and O'Brien (1974) and Pope (1982). Because the latter employ a Monte Carlo
solution, they appear to lack some conceptual and practical
advantages which the "discretised-PDF" nature of MFM offers.
However, given unlimited computer time and care to employ
precisely the same micro-mixing formulae, MFM and PDF-transport
should produce the same answers.
- There exist Kolmogorov-type models which do not employ
the enlarged-viscosity idea. These are the Reynolds-stress 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 Reynolds-Stress model could predict the "un-mixing" behaviour
reported in section 3.
- 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 Kolmogorov-type models, it could now
equally well be used for testing and augmenting the micro-mixing
hypotheses of MFM.
The argument presented in the foregoing lecture will now be
summarised, as follows:
- Of the two main approaches to turbulence modelling, namely
it is the second
which is better able to represent physical reality.
- Boussinesq-Kolmogorov (i.e. "enlarged-viscosity") and
- Reynolds-Prandtl (i.e. "intermingling-fragments"),
- Reynolds-Stress models, although they do not use the Boussinesq
enlarged-viscosity notion, are no better able to provide the needed
- PDF-transport models based upon Monte Carlo methods, although
directed at the right target, have built-in limitations which will
continue to prevent their widespread use.
- 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' intermingling-fragments 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 Twenty-First Century.
- 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 turbulence-energy
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 1-2, p56
- BE Launder, GJ Reece, W Rodi (1975) "Progress in the development of a
Reynolds-Stress 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 719-729
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 136-139, 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
13th International Symposium on Combustion, pp 649-657
The Combustion Institute
- DB Spalding (1987) "A turbulence model for buoyant and combusting
flows"; International J. for Numerical Methods in Engineering
vol 24, pp 1-23
- DB Spalding (1994) Poster session, International Heat Transfer
Conference, Brighton, England
- DB Spalding (1996) "Multi-fluid 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
Turbulence-modelling high-lights through four half-centuries
- Nineteenth century, second half
- Boussinesq (1877) proposes the "enlarged-viscosity" approach:
"A turbulent fluid is like a thick soup"
- Reynolds (1874) introduces the "inter-mingling-fragments" approach:
"A turbulent fluid is more like a stew"
(They did not actually use those words)
- Twentieth century, first half
- Prandtl (1925) uses the "inter-mingling-fragments" approach
(for his "mixing-length hypothesis"), but, for want of other
mathematical tools, casts the result in "enlarged-viscosity"
- 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
- 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.
- CFD-for-combustion specialists are forced to adopt the
"intermingling" fragments approach in order to fit the facts.
- CFD-for-multi-phase-flow specialists show how the multi-fluid
transport equations can be solved .
- Computers increase in power; but most turbulence modellers continue
to use power-restricted concepts.
- Nevertheless the first steps are taken to develop multi-fluid
models which make use of both the Boussinesq and Reynolds
- Twenty-first 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 general-purpose CFD code (which can be
Internet) has MFM built into it,
- the superior plausibility of MFM for many processes has already been
it is time at last to switch from Boussinesq to Reynolds.
- Thereafter, ?????
Links to explanations of the micro-mixing hypothesis of MFM