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### 7.6.Turbulence Modulation due to Particles

Several extensions of the KE-EP model for carrier-phase turbulence
modulation have been proposed in the literature, with notable
contributions from Pourahmadi and Humphrey [1983], Elghobashi et al
[1984], Chen et al [1985, 1986, 1990] and Mostafa and Mongia [1988].
None of these models can truly be recommended as a standard model;
so PHOENICS provides two representative extensions which allow for
these effects.

These are the models of Chen & Wood [1985,1986] and Mostafa & Mongia
[1988], which introduce additional source terms into the turbulence
transport equations. These source terms, which are coded in
subroutine GXDISP, are always negative and hence act to reduce KE
and EP. However, depending on the relative extent of the reductions
KE and EP, the turbulent viscosity may be either reduced or
increased by the presence of particles.

### Chen-Wood Model

The modifications proposed by Chen and Wood involve the introduction
the following additional volumetric source terms in the KE and EP
equations:

SKE,p = - (2*RHOd*Rc*Rd*KE/TP)*(1.-EXP(-BK*TP/TE)) .... (7.6.1)

SEP,p = - 2.*RHOd*Rc*Rd*EP/TP .... (7.6.2)

where: RHOd is the density of the dispersed phase; Rc and Rd are
the volume fractions of the carrier and dispersed phases; and BK is
empirical constant (=0.0825).

TP and TE are, respectively, timescales characterising the particle
response and large-scale turbulent motion:

TP = RHOd * Vslip * Rd / F" .... (7.6.3)

TE = CT*KE/EP .... (7.6.4)

where CT is an empirical constant (=0.165) and F"' is the absolute
magnitude of the interphase drag force per unit volume.

The foregoing source terms are activated by the following PIL
command:
PATCH(KEDISP,CELL,1,NX,1,NY,1,NZ,1,LSTEP)
COVAL(KEDISP,KE,GRND2,0.0); COVAL(KEDISP,EP,GRND2,0.0)

In addition, when STORE(TE,TP) appears in the Q1 file, the
timescales TE and TP may be printed in the RESULT file or viewed
via PHOTON or AUTOPLOT.

### Mostafa-Mongia Model

For the Mostafa-Mongia model, the additional volumetric source terms
in the KE and EP equations are:

SKE,p = - 2*RHOd*Rc*Rd*KE/(TE+TP) .... (7.6.5)

SEP,p = - 2*CPE*RHOd*Rc*Rd*EP/(TE+TP) .... (7.6.6)

where CPE is an empirical constant (=1.0), and TP and TE are given
by equations (6.3) and (6.4) above with CT=0.35.

The foregoing source terms are activated by the following PIL
command:
PATCH(KEDISP,CELL,1,NX,1,NY,1,NZ,1,LSTEP)
COVAL(KEDISP,KE,GRND1,0.0); COVAL(KEDISP,EP,GRND1,0.0)

As with the Chen-Wood model, TE and TP may be STOREd for printing in
the RESULT file and viewing via PHOTON and AUTOPLOT.

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