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7.4.Two-Equation K-E Turbulence Model

Since the turbulence is assumed to be a property of the first phase, modelled transport equations for the turbulent kinetic energy KE and its dissipation rate EP are solved only for this phase. For 2-phase high-Reynolds-number turbulent flows, the standard form of the KE-EP model may be summarised as follows, with ,t denoting differenti- ation with respect to time and ,i denoting differentiation with respect to distance:

(RHO1*R1*KE),t + (RHO1*R1*U1i*KE),i = RHO1*R1*(Pk-EP) + (RHO1*R1*{ENUL+ENUT/PRT(KE)}*KE,i),i + (RHO1*{ENUT/PRT(R1)}*KE*R1,i),i .... (7.4.1)

(RHO1*R1*EP),t + (RHO1*R1*U1i*EP),i = (RHO1*R1*EP/KE)*(C1*Pk-C2*EP + (RHO1*R1*{ENUL+ENUT/PRT(EP)}*EP,i),i + (RHO1*{ENUT/PRT(R1)}*EP*R1,i),i .... (7.4.2)

ENUT = CMUCD*KE**2/EP .... (7.4.3)

In the foregoing, RHO1, R1 and U1 are respectively: the density, volume fraction and velocity of phase 1 (the continuous phase); and Pk is the volumetric production rate of KE by shear forces:

Pk = ENUT*(Ui,j+Uj,i) Ui,j .... (7.4.4)

The following constants are used:

PRT(KE)=1.0, PRT(EP)=1.314, CMUCD=0.09, C1=1.44, C2=1.92.

The model presented above is applicable only in regions where the turbulence Reynolds number is high. Near walls, where the Reynolds number tends to zero, the model requires the application of so- called 'wall functions'. Following other workers, standard single- phase wall functions are employed in PHOENICS as described under the Encyclopaedia entry 'WALL-Functions'.

It should be mentioned that the WALL command assumes that phase 1 is the continuous phase, and will generate wall functions only for this phase. If wall effects are required for phase 2, such as for the phase-2 velocity variables, these have to be explicitly added in by the user.