Click here for an extract from an old lecture on 2FM applied to the environment

Click here for an old lecture on 2FM applied to combustion

- The general idea
- Alternative ways of distinguishing the two fluids
- The mathematical problem
- The physics of two-fluid turbulence
- Examples of auxiliary relations
- Determination of the constants
- Advantages of the two-fluid model
- Disadvantages of the two-fluid model
- Implementation in PHOENICS
- The future of the two-fluid model
- Sources of further information

Several authors have advocated thinking of a turbulent flow as a mixture of TWO fluids, each moving semi-independently in the same space.

The idea formed a part of the thinking of Reynolds (1874) and Prandtl (1925) as they considered how mass, momentum and energy were transported in turbulent fluids; and more recent promoters of the idea include: Spiegel (1972), Libby (1975), Dopazo(1977), Bray(1981), Spalding(1982) and Kollmann (1983).

The notion is easiest to understand (and perhaps of most immediate use) when the two fluids are chemically different, for example cold unburned gas and hot combustion products.

Each fluid is supposed to possess, at each position in space and time, its own velocity components, temperature, composition variables, volume fractions and (perhaps) pressure.

The volume fractions can be regarded as "probabilities of presence".

The two fluids can be distinguished in many ways, all of them being arbitrary.

Reynolds and Prandtl distinguished them by direction, fluid moving towards a surface being supposed to have different properties (eg along-surface momentum) from the fluid moving away from it.

For flows in the atmosphere, distinguishing upward-moving from downward-moving air is useful; for the temperature of the one is often markedly different from that of the other.

In a flame, the two fluids may comprise fully-burned gas on the one hand, and fully-unburned gas on the other.

In coastal waters, the two fluids may be distinguished by their salinity.

The mathematical problem presented by a two-fluid turbulence model is similar to that of predicting the behaviour of a two-phase flow such as a mixture of steam and water. Specifically, TWO sets of differential equations must be solved at the same time; moreover these equation sets are coupled, because:

- mass lost by one fluid is gained by the other;
- heat lost by one fluid is gained by the other;
- momentum lost by one is gained by the other;
- the same is true (apart from chemical reaction at the fluid- fragment boundaries) of transfer of chemical species.

Quantitative expressions are needed for the rates of exchange of mass, heat, momentum and chemical species between the fluid fragments; and these must express the physical processes of tearing, folding, inter-diffusion and separation which occur when fragments of the two fluids collide and mingle.

These auxiliary relations may express experimental observations; or they may be drawn from detailed fine-scale CFD studies of the interactions; or they may be guesses, shaped by dimensional analysis.

Whatever their origin, they are to this extent artificial: in truth there are NO sharp boundaries separating the one fluid from the other. There is an ANALOGY with two-phase flows, but no exact correspondence.

Examples of expressions which have been used in two-fluid models now follow:

- mass transfer: mdot = Cm * den * r1 * r2 * (0.5-ri) * |v1-v2|/l
- friction: fric = Cf * den * r1 * r2 * |v1-v2|/l
- shear source SHSOv= Cs * den * |v1-v2| * dw/dy
- viscosity: vist = Cvis * dens * r1 * r2 * l * |v1-v2|

where: Cm, Cf, Cs & Cvis are empirical constants; r1 & r2 are volume fractions of the 2 fluids; v1 & v2 are cross-stream velocities; w1 & w2 are stream-wise velocities; l = length scale; den = density.

Analysis of experimental data by means of the two-fluid model has resulted in a consensus regarding the constants, namely:

Cm = 10.0 ; Cf= 0.0375 ; Cs = 0.25 or 0.175 ; Cvis = 15.0

However, these are meaningful only when values are ascribed to the length scale, l. This, according to some recommendations, is taken to equal the Prandtl mixing length, calculated in the usual way; or it may be computed from an additional differential equation, of the form:

Dl/Dt = Ca.|v1 - v2| - Cb.S.l**2 where Ca & Cb are constants & S is the shear rate, and Ca = 0.03 Cb = 0.01

[Data from N Fueyo and DB Spalding, 1995]

- Like the Reynolds-stress models, two-fluid models are free
from the restrictions (and lack of realism) of the turbulent-
viscosity concept. They can predict "counter-gradient diffusion".
- Further (and unlike any single-fluid model), they can portray
adequately the interactions of pressure gradients and density
fluctuations which are major sources of generation of turbulent
motion.
- Two-fluid turbulence models allow proper account to be taken of the large differences and steep gradients of temperature and concentration which are present within turbulent reacting gases; so they can in principle and practice simulate flames realistically.

- The number of equations being twice as large as for single-
fluid models, computer times are somewhat longer. Moreover,
users of computer codes restricted to a single equation set
are often reluctant to change to one (such as PHOENICS) which
can handle two sets simultaneously.
This is probably the main reason that the two-fluid model has not yet become popular.

- The two-fluid concept is found hard to grasp by some; indeed
there are real conceptual difficulties, for example: What is
the best characteristic to use for distinguishing the two
fluids? Temperature? Velocity? Vorticity? Density? ....
- Knowledge of the fragment-interaction rates, and of what governs fragment size, is far from adequate. Much more research is needed.

The two-phase-option library contains the following list of cases:

Although only one thermodynamic phase is involved in these cases, they appear in the two-phase option, because they make use of the coding which was first introduced for two-phase flow.ONE-PHASE FLOWS computed by TWO-PHASE METHODS Case no. Two-fluid turbulence model; chemically inert Couette flow with buoyancy W975 Backward-facing step using two-fluid model W976 Mixing in a duct W974 Two-fluid turbulence model; Reacting flow Ducted flame using two-fluid model W977 1D piston-cylinder combustion W978 1D shock-induced propagation & detonation W979 Flame spread in plane channel W980

Extracts from the Q1 file of case W975 now follow:

GROUP 7. Variables (including porosities) named, stored & solved ONEPHS=F; SOLVE(P1,V1,V2,W1,W2,R1,R2,H1,H2) SOLUTN(C1,Y,Y,P,P,P,P); SOLUTN(C2,Y,Y,P,P,P,P) SOLUTN(C3,Y,Y,P,P,P,P); SOLUTN(C4,Y,Y,P,P,P,P) SOLUTN(C5,Y,Y,P,P,P,P); SOLUTN(C6,Y,Y,P,P,P,P) INTMDT=22;LEN1=23;VIST=24;NAME(INTMDT)=MDOT;NAME(LEN1)=LEN NAME(INTMDT)=MDOT;NAME(LEN1)=LEN;NAME(VIST)=VIS SOLUTN(MDOT,Y,N,N,N,N,N); SOLUTN(VIST,Y,N,N,N,N,N) SOLUTN(LEN1,Y,N,N,N,N,N)GROUP 10. Interphase-transfer processes and properties. CFIPS=GRND4;CFIPA=0.0;CFIPB=1.0;CFIPD=-1.0;CFIPC=0.05 CMDOT=GRND1;CMDTA=10.0;CMDTB=0.5;CMDTC=0.0 CINT(C1)=10.0;CINT(C2)=10.0;CINT(C5)=0.1;CINT(C6)=0.1 CINT(C3)=0.0;CINT(C4)=0.0

GROUP 13. Boundary conditions, and special sources. SOUTH WALL REAL(COEF);COEF=1.0 PATCH(FIXED,SOUTH,1,1,1,1,1,NZ,1,1) COVAL(FIXED,W1,FIXVAL,0.0); COVAL(FIXED,H1,1.0,1.0) COVAL(FIXED,C1,COEF,1.0); COVAL(FIXED,C5,COEF,1.0) COVAL(FIXED,C3,COEF,0.0); COVAL(FIXED,C4,COEF,0.0) NORTH WALL PATCH(MOVING,NORTH,1,1,NY,NY,1,NZ,1,1) COVAL(MOVING,W2,FIXVAL,2.0*CHARW); COVAL(MOVING,H2,1.0,0.0) COVAL(MOVING,C2,COEF,0.0); COVAL(MOVING,C6,COEF,0.0) COVAL(MOVING,C3,COEF,0.0); COVAL(MOVING,C4,COEF,0.0)

WHOLE FIELD PATCH(SHSOURCE,CELL,1,1,1,NY,1,NZ,1,1) COVAL(SHSOURCE,V1,FIXFLU,GRND5); COVAL(SHSOURCE,V2,FIXFLU,GRND5) SHSOA=1.E0 BODY FORCE PATCH(BUOY,PHASEM,1,1,1,NY,1,NZ,1,1) COVAL(BUOY,V1,FIXFLU,GRND4) IBUOYB=14;IBUOYC=15;CSG2=BUOY;BUOYA=0.0;BUOYD=0.025 LENGTH-SCALE SOURCE PATCH(LESO,PHASEM,1,1,1,NY,1,NZ,1,1) COVAL(LESO,C3,FIXFLU,GRND1); COVAL(LESO,C4,FIXFLU,GRND1) ELSOA=0.025

1.00 M....+....+....+....+....+....+....+....+....+.A..D ^ .A . | 0.90 + An extract from the results: A + max . profiles across the channel. DC 0.80 + A + . <---- fixed wall moving wall----> D . 0.70 + U C+ . A M . 0.60 + U U A M C U legend . A B B B B B B B B B B B 0.50 +M D U U U U U U U U U U D C M+ A=V1 . D M M M M M M M M M M M M . B=V2 0.40 U C + C=W1 . D B B U . D=W2 0.30 +D M U + U=R1 . M . M=MDOT 0.20 + B + DC B. min 0.10 + + | . B -----------------> distance Y . | 0.00 CB...+....+....+....+....+....+....+....+....+....M V

Although slow to gain popularity, the two-fluid concept is becoming better understood, especially as other CFD-code vendors introduce IPSA into their software packages.

In one respect, the two-fluid model first incorporated into PHOENICS may be over-elaborate; for the relative velocities of the fluids are often small enough to be neglected (with EQUVEL=T), or computed by way of an algebraic-slip approximation.

The latter formulation permits extension to the treatment of the relative motion of more fluids than two, which is what is needed for greater realism.

It may therefore be that the two-fluid model will before long give way to MULTI-fluid models, to which however it forms a useful and educative introduction.

Models comprising up to one hundred fluids have already been used with the latest version of PHOENICS.

- Some PhD theses concerned with two-fluid turbulence modelling
by researchers at the Computational Fluid Dynamics Unit of
Imperial College, London University:
MJ Andrews (1986) "Turbulent mixing by Rayleigh-Taylor instability"

N Fueyo (1992) "Two-fluid models of turbulence for axi-symmetrical jets and sprays"

MR Malin (1986) "Turbulence modelling for flow and heat transfer in jets, wakes and plumes"

ST Xi (1986) "Transient turbulent jets of miscible and immiscible fluids"

- Reports and publications:
DB Spalding (1987) "A turbulence model for buoyant and
combusting flows" Int J for Num Methods in Engg, v 24, p1
JZ Wu (1987) "The application of the two-fluid model of turbulence to ducted flames" IC CFDU Report, June 1987

JO Ilegbusi and DB Spalding (1987) "A two-fluid model of turbulence and its application to near-wall flows" I J PhysicoChemical Hydrodynamics , vol 9, pp 127-160

JO Ilegbusi and DB Spalding (1987) "Application of a two-fluid model of turbulence to turbulent flows in conduits and shear layers" I J PhysicoChemical Hydrodynamics, vol 9, pp 161-181

DB Spalding (1993) Lecture delivered at "First International Conference on Air Pollution", Monterrey, Mexico. The text is included in "Lecture on the two-fluid turbulence model" supplied with the PHOENICS software package

DB Spalding (1995) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland

DB Spalding (1995) "Multi-fluid models of turbulent combustion"; CTAC95 Conference, Melbourne, Australia

DB Spalding (1995) "Multi-fluid models of turbulence" PHOENICS User Conference, Trento, Italy

D Freeman and DB Spalding (1995) "The multi-fluid turbulent combustion model and its application to the simulation of gas explosions"; The PHOENICS Journal