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
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:
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:
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]
This is probably the main reason that the two-fluid model has not yet become popular.
The two-phase-option library contains the following list of cases:
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
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.
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.
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"
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