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9.7.1 Euler Equations

For inviscid flows, FLUENT solves the Euler equations. The mass conservation equation is the same as for a laminar flow, but the momentum and energy conservation equations are reduced due to the absence of molecular diffusion.

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

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 (9.7-1)

Equation  9.7-1 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 (9.7-2)

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 is described by

 \frac{\partial}{\partial t} (\rho {\vec v}) + \nabla \cdot (\rho {\vec v}{\vec v}) = -\nabla p + \rho {\vec g} + {\vec F} (9.7-3)

where $p$ is the static pressure and $\rho \vec{g}$ and $\vec{F}$ are the gravitational body force and external body forces (e.g., forces 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.

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

 \frac{\partial}{\partial t} (\rho v_x) + \frac{1}{r}\frac{\p... ...al r} (r \rho v_r v_x) = - \frac{\partial p}{\partial x} + F_x (9.7-4)


 \frac{\partial}{\partial t} (\rho v_r) + \frac{1}{r}\frac{\p... ...al r} (r \rho v_r v_r) = - \frac{\partial p}{\partial r} + F_r (9.7-5)


 \nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_r}{\partial r} + \frac{v_r}{r} (9.7-6)

Energy Conservation Equation

Conservation of energy is described by

 \frac{\partial}{\partial t} (\rho E) + \nabla \cdot ({\vec v... ...rho E + p)) = -\nabla \cdot \left(\sum_j h_j J_j \right) + S_h (9.7-7)

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