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7.3.Turbulent Diffusion of Phase Mass

7.3.1. Introduction

In the PHOENICS IPSA option (ONEPHS=F), the phase-diffusion term represents the turbulent flux associated with correlations between fluctuating velocity and volume fraction.

The flux is modelled via a gradient-diffusion approximation, and it accounts for diffusion of the dispersed phase due to turbulence in the continuous phase.

7.3.2.Description and Equations

Phase-diffusion terms are present in the phase continuity equations;

    (r_k*R_k)_,t + (r_k*R_k*Vk)_,i = [r_k*G_k*R_k,i]_,i      .... (7.3.1)

and the phase-F equations;

   (r_k*R_k*F_k),t + (r_k*R_k*V_k*F_k)_,i = [r_k*G_Fk*R_k*F_k,i]_,i + [r_k*G_k*F_k*R_k,i]_,i      .... (7.3.2)

Here r_k, R_k and V_k denote the density, volume fraction and velocity vector of phase k; G_k is the diffusion coefficient for the volume fractions; and G_Fk denotes the conventional diffusion coefficient of the dependent variable F for phase k;

   G_Fk = ( n_l/s_l,Fk + n_t/s_t,Fk )                  .... (7.3.3)

where n_l and n_t are the reference laminar and turbulent kinematic viscosities; and s_l,Fk and s_t,Fk are the laminar and turbulent Prandtl numbers for F_k.

The phase-diffusion coefficient G_k is calculated from:

   G_k = n_t/s_t,k                                      .... (7.3.4)

where s_t,k is the turbulent Prandtl number for the volume fraction of phase k, which is taken as unity by default.

The turbulent Prandtl numbers are defined by the PIL variables PRT(R1) and PRT(R2), and their equality is enforced by PHOENICS for reasons of volume conservation. Phase-diffusion fluxes also appear in the continuity equation for the shadow volume fraction RS, and so PHOENICS also ensures PRT(RS)=PRT(R2).

The finite-volume analogues of the phase-diffusion fluxes integrated over the control volume are given in 'The PHOENICS Equations',TR/99, 1987 (p11.1-11.2,p7.1-7.2,p6.1) for regular coordinates. For non-orthogonal staggered-mesh body-fitted coordinate systems, the phase diffusion fluxes through each cell face for a given phase are computed from:

    D_e = (G_e/d_e)*[Me_pe*(r_P-r_E) + Me_pn*(A_n/A_e)*(r_P-r_N) + Me_ph*(A_h/A_e)*(r_P-r_H) ]    ...(7.3.5)

    D_n = (G_n/d_n)*[Mn_pn*(r_P-r_N) + Mn_pe*(A_e/A_n)*(r_P-r_E) + Mn_ph*(A_h/A_n)*(r_P-r_H) ]    ...(7.3.6)

    D_h = (G_h/d_h)*[Mh_ph*(r_P-r_H) + Mh_pe*(A_e/A_h)*(r_P-r_E) + Mh_pn*(A_n/A_h)*(r_P-r_N) ]    ...(7.3.7)

where the A's are projected areas, the d's are resolutes of the internodal distance, and the M's are the geometric coefficients employed in computing the corresponding convection fluxes through the cell face. For orthogonal coordinate systems, only the first term in each of the foregoing equations is present, because the geometric coefficients are zero for all other terms. 

7.3.3. Miscellaneous Options

When STORE(PHDE,PHDN,PHDH) appears in the Q1 file, the 1st phase diffusion fluxes through the east, north and high cell faces are stored whole field and printed to the RESULT and to the PHI file for viewing by the PHOENICS graphical post-processors. The printed fluxes are integrated over the contol volume and the convention is that they are positive if the flux is in the coordinate direction.

Phase-diffusion effects can be removed entirely, by setting the Prandtl numbers to large values, i.e. PRT(R1)=PRT(R2)=1.E10, or alternatively by introducing the PIL commands:

TERMS(R1,P,P,N,P,P,P);TERMS(R2,P,P,N,P,P,P)

in the Q1 input file.

It may be desirable to remove phase diffusion at pre-selected grid lines or surfaces, e.g. those aligned with free surfaces. This can be done by introducing an appropriate GP12 PATCH in the Q1 input file. For example:

PATCH(GP12DFHA,CELL,IXF,IXL,IYF,IYL,IZF,IZL,ITF,ITL) COVAL(GP12DFHA,R1,ZERO,ZERO); COVAL(GP12DFHA,R2,ZERO,ZERO)

would cut out the high-face phase-diffusion fluxes for R1 and R2 in the cells and for the duration indicated by arguments 3 to 10 of the PATCH command. The high-face diffusion fluxes for R1 and R2 are multiplied by the third argument (i.e.zero ) of the above COVAL statements ( see PHENC 'Group 12 Features' for more details ).

Finally, if the User sets UDIFF=T in the Q1 input file, the phase-diffusion fluxes may be accessed directly for modification in Group 8 Section 9 of GROUND. For example, the following coding sequence accesses and prints out the phase-diffusion fluxes for R2:

         IF(INDVAR.EQ.R2) THEN
           IF(NDIREC.EQ.5) THEN
             CALL PRN('DIFH',LD11)
           ELSE IF(NDIREC.EQ.3) THEN
             CALL PRN('DIFW',LAW)
             CALL PRN('DIFE',LAE)
           ELSE IF(NDIREC.EQ.1) THEN
             CALL PRN('DIFS',LAS)
             CALL PRN('DIFN',LAN)
           ENDIF
         ENDIF

The block-location indices for the phase-diffusion fluxes are LAE for east, LAW for west, LAN for north, LAS for south, and LD11 for high. PHOENICS enforces reciprocity of the low and high fluxes and any GP12 modifications will be made after the above coding.

mrm 10/05/02