Equation in Burns.C of Turbulent dispersion model of Burns et al.
Burns is one of the turbulent dispersion models of interficial models in reactingEulerFoam. And the reference is
Burns, A. D., Frank, T., Hamill, I., & Shi, J. M. (2004, May).The Favre averaged drag model for turbulent dispersion in Eulerian multi-phase flows.In 5th international conference on multiphase flow, ICMF (Vol. 4, pp. 1-17). Reference link
The equation used in Burns.C
from the reference is
\vec M_{\alpha}^{TD}=-\vec M_{\beta}^{TD}=-\frac{3}{4}C_D\frac{\overline r_{\beta}\rho_{\alpha}}{d_\beta}|\vec{U_\beta}-\vec{U_\alpha}|\frac{\nu_{t\alpha}}{\sigma_{r\alpha}}(\frac{1}{\overline r_{\alpha}}+\frac{1}{\overline r_{\beta}})\nabla \overline r_{\alpha}
And the correspponding source code in Burns.C
is
0.75
*drag.CdRe()
*pair_.continuous().nu()
*continuousTurbulence().nut()
/(
sigma_
*sqr(pair_.dispersed().d())
)
*pair_.continuous().rho()
*pair_.dispersed()
*(
1.0/max(pair_.dispersed(), residualAlpha_)
+ 1.0/max(pair_.continuous(), residualAlpha_)
);
I tried to translate it into the mathematical form as
\frac{3}{4}C_D\frac{\nu_\alpha\rho_\alpha \overline r_\beta}{d^2_\beta}\frac{\nu_{t\alpha}}{\sigma_{r\alpha}}(\frac{1}{\overline r_{\alpha}}+\frac{1}{\overline r_{\beta}})
The two equations seem not to be the same. The interphase velocity term |\vec{U_\beta}-\vec{U_\alpha}|
is missed in the source code and there is an additional term \frac{\nu_{\alpha}}{d_{\beta}}
in it.
Is there any relationship between these two terms?