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9.2 Continuity and Momentum Equations

For all flows, FLUENT solves conservation equations for mass and momentum. For flows involving heat transfer or compressibility, an additional equation for energy conservation is solved. For flows involving species mixing or reactions, a species conservation equation is solved or, if the non-premixed combustion model is used, conservation equations for the mixture fraction and its variance are solved. Additional transport equations are also solved when the flow is turbulent.

In this section, the conservation equations for laminar flow in an inertial (non-accelerating) reference frame are presented. The equations that are applicable to rotating reference frames are presented in Chapter  10. The conservation equations relevant to heat transfer, turbulence modeling, and species transport will be discussed in the chapters where those models are described.

The Euler equations solved for inviscid flow are presented in Section  9.7.

The Mass Conservation Equation

The equation for conservation of mass, or continuity equation, can be written as follows:

 \frac{\partial \rho}{\partial t} + {\bf {\nabla}} \cdot (\rho {\vec v}) = S_m (9.2-1)

Equation  9.2-1 is the general form of the mass conservation equation and is valid for incompressible as well as compressible flows. The source $S_m$ is the mass added to the continuous phase from the dispersed second phase (e.g., due to vaporization of liquid droplets) and any user-defined sources.

For 2D axisymmetric geometries, the continuity equation is given by

 \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial ... ...ac{\partial}{\partial r} (\rho v_r) + \frac{\rho v_r}{r} = S_m (9.2-2)

where $x$ is the axial coordinate, $r$ is the radial coordinate, $v_x$ is the axial velocity, and $v_r$ is the radial velocity.

Momentum Conservation Equations

Conservation of momentum in an inertial (non-accelerating) reference frame is described by [ 24]

 \frac{\partial}{\partial t} (\rho {\vec v}) + \nabla \cdot (... ... \cdot (\overline{\overline{\tau}}) + \rho {\vec g} + {\vec F} (9.2-3)

where $p$ is the static pressure, $\overline{\overline{\tau}}$ is the stress tensor (described below), and $\rho {\vec g}$ and ${\vec F}$ are the gravitational body force and external body forces (e.g., that arise from interaction with the dispersed phase), respectively. ${\vec F}$ also contains other model-dependent source terms such as porous-media and user-defined sources.

The stress tensor $\overline{\overline{\tau}}$ is given by

 \overline{\overline{\tau}} = \mu \left[ (\nabla {\vec v} + \... ...vec v}\;^{\rm T}) - \frac{2}{3} \nabla \cdot {\vec v} I\right] (9.2-4)

where $\mu$ is the molecular viscosity, $I$ is the unit tensor, and the second term on the right hand side is the effect of volume dilation.

For 2D axisymmetric geometries, the axial and radial momentum conservation equations are given by

$\displaystyle \frac{\partial}{\partial t} (\rho v_x) + \frac{1}{r}\frac{\partia... ...x}{\partial x} - \frac{2}{3} \left(\nabla \cdot \vec{v} \right) \right) \right]$      
$\displaystyle + \frac{1}{r}\frac{\partial}{\partial r} \left[ r \mu \left(\frac{\partial v_x}{\partial r} + \frac{\partial v_r}{\partial x} \right) \right] + F_x$      
$\displaystyle \;$     (9.2-5)


\frac{\partial}{\partial t} (\rho v_r) + \frac{1}{r}\frac{\pa... ...}{\partial x} + \frac{\partial v_x}{\partial r} \right) \right]

 + \frac{1}{r}\frac{\partial}{\partial r} \left[ r \mu \left... ...left(\nabla \cdot \vec{v} \right) + \rho \frac{v_z^2}{r} + F_r (9.2-6)


 \nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_r}{\partial r} + \frac{v_r}{r} (9.2-7)

and $v_z$ is the swirl velocity. (See Section  9.5 for information about modeling axisymmetric swirl.)

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© Fluent Inc. 2006-09-20