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25.5.1 Governing Equations in Vector Form

The system of governing equations for a single-component fluid, written to describe the mean flow properties, is cast in integral Cartesian form for an arbitrary control volume $V$ with differential surface area $d{\mbox{\boldmath$A$}}$ as follows:


 \frac{\partial }{\partial t} \int_V {\mbox{\boldmath$W$}}\, ... ...t d{{\mbox{\boldmath$A$}}} = \int_V {\mbox{\boldmath$H$}}\, dV (25.5-1)

where the vectors ${\mbox{\boldmath$W$}}, {\mbox{\boldmath$F$}}$, and ${\mbox{\boldmath$G$}}$ are defined as

 {{\mbox{\boldmath $W$}}} = \left\{ \begin{array}{@{}l@{}} \... ...$\tau$}}_{ij} v_j + {\mbox{\boldmath $q$}} \end{array}\right\}

and the vector ${\mbox{\boldmath$H$}}$ contains source terms such as body forces and energy sources.

Here $\rho$, ${\mbox{\boldmath$v$}}$, $E$, and $p$ are the density, velocity, total energy per unit mass, and pressure of the fluid, respectively. ${\mbox{\boldmath$\tau$}}$ is the viscous stress tensor, and ${\mbox{\boldmath$q$}}$ is the heat flux.

Total energy $E$ is related to the total enthalpy $H$ by


 E = H - p/\rho (25.5-2)

where


 H = h + \vert{\mbox{\boldmath$v$}}\vert^2/2 (25.5-3)

The Navier-Stokes equations as expressed in Equation  25.5-1 become (numerically) very stiff at low Mach number due to the disparity between the fluid velocity ${\mbox{\boldmath$v$}}$ and the acoustic speed $c$ (speed of sound). This is also true for incompressible flows, regardless of the fluid velocity, because acoustic waves travel infinitely fast in an incompressible fluid (speed of sound is infinite). The numerical stiffness of the equations under these conditions results in poor convergence rates. This difficulty is overcome in FLUENT's density-based solver by employing a technique called (time-derivative) preconditioning [ 396].


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