Favre averaged Navier-Stokes equations
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(Difference between revisions)
Line 6: | Line 6: | ||
\frac{\partial \rho}{\partial t} + | \frac{\partial \rho}{\partial t} + | ||
\frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0 | \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0 | ||
- | </math> | + | </math> (1) |
- | + | ||
- | <math> | + | :<math> |
\frac{\partial}{\partial t}\left( \rho u_i \right) + | \frac{\partial}{\partial t}\left( \rho u_i \right) + | ||
\frac{\partial}{\partial x_j} | \frac{\partial}{\partial x_j} | ||
\left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0 | \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0 | ||
- | </math> | + | </math> (2) |
- | <math> | + | :<math> |
\frac{\partial}{\partial t}\left( \rho e_0 \right) + | \frac{\partial}{\partial t}\left( \rho e_0 \right) + | ||
\frac{\partial}{\partial x_j} | \frac{\partial}{\partial x_j} | ||
\left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0 | \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0 | ||
- | </math> | + | </math> (3) |
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous | For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous |
Revision as of 08:09, 5 September 2005
The instantaneous continuity equation, momentum equation and energy equation for a compressible fluid can be written as:
- (1)
- (2)
- (3)
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
Where the trace-less viscous strain-rate is defined by:
The heat-flux, , is given by Fourier's law:
Where the laminar Prandtl number is defined by:
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
Where and are constant.
The total energy is defined by:
Note that the corresponding expression~\ref{eq:fav_total_energy} for Favre averaged turbulent flows contains an extra term related to the turbulent energy.