Encyclopaedia Index

Turbulent mixing and chemical reaction;
the multi-fluid approach
1995-2010

by

Brian Spalding, CHAM Ltd, London, England

Historical notes.
(1) The original of this document is a lecture delivered in Moscow, in May 1997, at The International Symposium on the Physics of Heat Transfer in Boiling and Condensation.
Its content was expanded for delivery in the Institute of Professor Kemal Hanjalic of the University of Delft during the following year; and further additions were made for later occasional deliveries, so that it is now too long to be delivered, in toto, in a single session anywhere at all.
Nevertheless, it has seemed best to preserve it as a record of the author's thoughts about MFM during the last years of the twentieth century.

(2) Until May 2010, the material of the lecture was distributed between many .htm and .gif files. Their contents have now been collected into a single document for ease of distribution; and the opportunity has been taken to augment it by addition of previously omitted material, namely

  1. that concerned with the two-dimensional fourteen-fluid model used already in 1996 for the simulation of a Bunsen-burner flame in section 5.5,
  2. appendix 2 concerned with the connection between MFM and the so-called 'flamelet' model of turbulent combustion,
  3. appendix 3 concerned with simulation of smoke generation, and
  4. appendix 4 concerned with gas-turbine combustion.

Abstract

Discretization of scalar-variable space, in the same manner as is customary for geometric space, makes possible the simulation of many turbulent single- and multi-phase flow phenomena for which conventional turbulence models fail, especially those influenced by body forces, or by chemical reaction.

This opportunity is exploited by the Multi-Fluid Model (MFM) of turbulence, which may be regarded as an extension and generalization of the "PDF-transport" model of Dopazo, O'Brien, Pope, et al. It is also the successor to, and generaliser of, numerous two- fluid models of the kind which were already envisaged by Reynolds and Prandtl.

MFM uses a conventional finite-volume method for computing the discretized PDFs, which may be one-, two- or multi-dimensional.

The lecture explains the nature and practical utility of MFM. Examples of its application to both chemically-inert and chemically-reactive flow phenomena are presented.

Contents of the lecture


  1. The task to be performed: computing the PDF
  2. Efforts to avoid computing the PDF
  3. Pioneering efforts to compute the PDFs
  4. The multi-fluid model (MFM) approach to PDFs
  5. First steps towards MFM for combustion
  6. Application of MFM to the ideal well-stirred reactor
  7. Application of MFM to 3D processes in engineering equipment
  8. The plane uniform-density mixing layer
  9. The future of MFM
  10. References
Appendix 1 : governing equations and underlying assumptions
Appendix 2 : Connections between the multi-fluid and flamelet models of turbulent combustion
Appendix 3 : The simulation of smoke generation in a 3-D combustor, by means of the multi-fluid model of turbulent chemical reaction
Appendix 4 : The use of CFD in the design and development of gas-turbine combustors

1. The task to be performed: computing the PDF

Computer simulation of many turbulent-flow phenomena, especially those influenced by body forces or chemical reactions, requires:

  1. quantitative description of the fluid state in terms of probability-density functions (PDFs), because knowledge of mixture-averages and of root-mean-square fluctuations is not enough;
  2. quantitative physical hypotheses for the heat-and-mass-transfer, micro-mixing, mechanical, chemical and other processes which tend to change the PDFs;
  3. incorporation of these hypotheses, together with the conservation laws of physics, into mathematical equations which are capable of being solved numerically;
  4. computation of the solutions to these equations, which then yield the required PDFs.

The following picture shows what is meant by a probability-density function.

The PDF is the curve on the left of the picture. The task is to calculate its shape.

On the right is a graphical reminder of the fact that a turbulent fluid consists of a random- seeming assembly of fluid fragments in various states.


A one-dimensional probability-density function, with probability- density plotted vertically and temperature horizontally.

The "spikes" on the left and right show that there are large amounts of extremely-cold and extremely-hot fluid.

The curve between them shows how much of the fluid is in the intermediate temperature ranges.

This information is needed for predicting, say:

Knowledge of the average temperature is of little or no use for these purposes.


2. Efforts to avoid performing the task: EBU, EDC, 2FM, other presumed PDF

In the past, it was believed (with some truth) that the computational task was too intractable to be performed.

This belief led to searches for labour-saving schemes, based upon the presumptions that:
either
the PDF has a known, simple shape;
or
certain statistical properties of the PDF will suffice.

Both of these will be discussed in this section.


2.1 Presuming the shape of the PDF

(a) The eddy-break-up model (EBU), (Spalding, 1971)

The eddy-break-up model for turbulent combustion represents the population distribution by two spikes, as shown:



              |

       mass   |   ^unburned

       fractions  |                

       of the |   |                

       two    |   |                ^burned

       gases  |   |                |

              |   |                |

              |   |                |

              |   |                |

              |___|________________|_____

                  0  reactedness-> 1

At any location, the gas mixture is supposed to comprise fully-burned and fully-unburned gases.

Their proportions vary with location in accordance with laws of convection, diffusion and source/sink interactions.

The latter represent the transformation of unburned to burned at a volumetric rate proportional to the local mean-flow shear rate.

In some variants, a chemical-kinetic rate-limitation was introduced.

(b) The eddy-dissipation concept (EDC), (Magnussen, 1976)

The eddy-dissipation concept model for turbulent combustion also presumes a two-spike population distribution, represented as shown:



              |

       mass   |         ^mean mixture

       fractions        |

       of the |         |

       two    |         |

       gases  |         |

              |         |

              |         |    ^fine

              |         |    |structures

              |___|_____|____|_____|__

                  0  reactedness-> 1

At any location, the gas mixture is supposed to comprise mean mixture and "fine structures",

Their proportions vary with location in accordance with laws of convection, diffusion and source/sink interactions.

The latter involve mass transfer between the two gases at a volumetric rate proportional to the local mean-flow shear rate.

Chemical kinetics controls the rate of fine-structure reaction and so the mean-mixture reactedness.

(c) The two-fluid model (2FM), (Spalding, 1987 )

The two-fluid model of turbulent combustion presumes a two-spike population distribution, represented as shown:



              |

       mass   |         ^slower fluid

       fractions        |

       of the |         |

       two    |         |

       gases  |         |

              |         |

              |         |    ^faster fluid

              |         |    |

              |___|_____|____|_____|__

                  0  reactedness-> 1

At any location, the gas mixture is supposed to comprise faster- and slower- moving gases.

Their speeds & proportions vary with location in accordance with laws of convection, diffusion and source/sink interactions.

The latter involve mass transfer between the two gases at a volumetric rate proportional to the local relative velocity.

Chemical kinetics controls the rate of chemical reaction within each gas, and so their individual reactednesses.

(d) Examples of two-fluid-model (2FM) simulations; the atmosphere.
A 2-fluid model applied to the environment

In the atmosphere, and in natural waters, gravitational forces have a major effect on the fluid-dynamic phenomena, precisely because of the fact that fragments of denser fluid (colder air or more-saline water) are mingled with lighter ones.

The latter tend to rise, and the former to fall, often with striking effects.

For example, hurricanes gather their destructive power from the fact that the upwardly moving moist-air fragments, as they rise to lower-pressure altitudes, shed much of their water content as rain.

The latent heat of condensation has the effect of increasing still further the disparity of density between the upward- and downward-moving fluids, the relevant motion of which is therefore maintained, despite the friction between them.

Two-fluid models of turbulence are well able to simulate such phenomena. For example, the present author showed [57] how PHOENICS could be used to calculate the variation with time of the upward- and downward-moving members of the "atmospheric population" as a consequence of the heating and cooling of the surface of the earth.

The following pictures represent what happens when a cool wind, moving from left to right, is heated as a consequence of contact with hot earth beneath it.



temperatures in upward-(left) and downward-(right) moving fluids

Not surprisingly, the upward-moving fluid has a higher temperature than the downward-moving fluid.



volume fraction of upward-moving fluid

The upward-moving fluid occupies more than 50% of space near the hot surface.


velocity vectors in upward-(left) and downward-(right) moving fluids



the effective viscosity of the mixture



contours of pollutant in upward-(left) and downward-(right) moving fluids

Some line-printer output from the PHOENICS library case W975 now follows; this concerns flow between a hot surface at rest and an upper cold moving surface. This is a Couette-flow idealization of an unstable atmosphere.

Each plot shows the variation with vertical distance of some properties of the two fluids, namely: