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23.4.5 Relative (Slip) Velocity and the Drift Velocity

The relative velocity (also referred to as the slip velocity) is defined as the velocity of a secondary phase ( $p$) relative to the velocity of the primary phase ( $q$):


 \vec{v}_{pq} = \vec{v}_p - \vec{v}_q (23.4-9)

The mass fraction for any phase ( $k$) is defined as


 c_k = \frac{\alpha_k \rho_k}{\rho_m} (23.4-10)

The drift velocity and the relative velocity ( $\vec{v}_{qp}$) are connected by the following expression:


 \vec{v}_{{\rm dr},p} = \vec{v}_{pq} - \sum_{k=1}^{n} c_k \vec{v}_{qk} (23.4-11)

FLUENT's mixture model makes use of an algebraic slip formulation. The basic assumption of the algebraic slip mixture model is that to prescribe an algebraic relation for the relative velocity, a local equilibrium between the phases should be reached over short spatial length scale. Following Manninen et al. [ 230], the form of the relative velocity is given by:


 \vec {v}_{pq} = \frac{\tau_p}{f_{\rm drag}} \frac{(\rho_p - \rho_m)}{\rho_p} \vec {a} % (23.4-12)

where $\tau_p$ is the particle relaxation time


 \tau_p = \frac{\rho_p d_p^2}{18 \mu_q} (23.4-13)

$d$ is the diameter of the particles (or droplets or bubbles) of secondary phase $p$, $\vec{a}$ is the secondary-phase particle's acceleration. The default drag function $f_{\rm drag}$ is taken from Schiller and Naumann [ 321]:


 f_{\rm drag} = \left\{ \begin{array}{ll} 1 + 0.15 \; {\rm Re... ...0 \\ 0.0183 \; {\rm Re} & {\rm Re} > 1000 \end{array} \right. (23.4-14)

and the acceleration $\vec{a}$ is of the form


 \vec{a} = \vec{g} - ( \vec{v}_m \cdot \nabla) \vec{v}_m- \frac{\partial \vec{v}_m}{\partial t} (23.4-15)

The simplest algebraic slip formulation is the so-called drift flux model, in which the acceleration of the particle is given by gravity and/or a centrifugal force and the particulate relaxation time is modified to take into account the presence of other particles.

In turbulent flows the relative velocity should contain a diffusion term due to the dispersion appearing in the momentum equation for the dispersed phase. FLUENT adds this dispersion to the relative velocity:


 \vec {v}_{pq} = \frac{(\rho_p - \rho_m) d^2_p}{18 \mu_q f_{\... ...\vec {a} - \frac{\nu_m}{\alpha_p \sigma_{D}} \nabla \alpha_q % (23.4-16)

where ( $\nu_m$) is the mixture turbulent viscosity and ( $\sigma_{D}$) is a Prandtl dispersion coefficient.

When you are solving a mixture multiphase calculation with slip velocity, you can directly prescribe formulations for the drag function. The following choices are available:

See Section  23.5.4 for more information on these drag functions and their formulations, and Section  23.11.1 for instructions on how to enable them.

Note that, if the slip velocity is not solved, the mixture model is reduced to a homogeneous multiphase model. In addition, the mixture model can be customized (using user-defined functions) to use a formulation other than the algebraic slip method for the slip velocity. See the separate UDF Manual for details.


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© Fluent Inc. 2006-09-20