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## Turbulent mixing and chemical reaction; the multi-fluid approach

### by

## Brian Spalding, CHAM Ltd, London, England

###
A lecture , 1998

###### Historical note.*The original of this lecture is one 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
Hanjalic of the University of Delft during the following year; and
further additions have been made for later occasional deliveries, so
that it is now too long to be delivered, in toto, in a single session
anywhere.

Nevertheless, it has seemed best to preserve it as a record of the
author's thoughts about MFM during 1997 and 1998.

### 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 are presented to both chemically-inert and
chemically-reactive flow phenomena.

### Contents of the lecture

- The task to be performed: computing the PDF
- Efforts to avoid computing the PDF
- EBU: the eddy-break-up model
- EDC: the eddy-dissipation concept
- 2FM: the two-fluid model
- Presuming the shape of the PDF
- Presuming that a few statistical properties will suffice

- Pioneering efforts to compute the PDFs
- The multi-fluid model (MFM) approach to PDFs
- First steps towards MFM for combustion:
the four-fluid model
- The flame-spread-in-a-plane-duct problem
- Transient flame spread in a baffled duct
- Explosion in an off-shore oil platform

- Application of MFM to the ideal well-stirred reactor
- With uniform fuel-air ratio; the 1D population
- With non-uniform fuel-air ratio; the 2D population
- Concluding remarks about the stirred-reactor results

- Application of MFM to 3D processes in engineering equipment
- Transient flow in a paddle-stirred reactor
- Smoke generation in a gas-turbine combustor
- Prediction of turbo-machinery flows

- The plane uniform-density mixing layer
- The problem and its solution
- Discussion of the results

- The future of MFM
- Concluding remarks about the nature of MFM
- Mathematical and computational tasks
- Tests of realism
- Conceptual developments

- References

Appendix: governing equations and underlying
assumptions