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25.4.4 Steady-State Iterative Algorithm

If you are performing a steady-state calculation, the governing equations for the pressure-based solver do not contain time-dependent terms. For steady-state flows,
Section  25.3 describes control-volume-based discretization of the steady-state transport equation (see Equation  25.2-1).

Under-Relaxation of Variables

The under-relaxation of variables is used in all cases for some material properties, in the NITA solver for solution variables, and in the pressure-based coupled algorithm where this explicit under-relaxation is used for momentum and pressure.

Because of the nonlinearity of the equation set being solved by FLUENT, it is necessary to control the change of $\phi$. This is typically achieved by under-relaxation of variables (also referred to as explicit relaxation), which reduces the change of $\phi$ produced during each iteration. In a simple form, the new value of the variable $\phi$ within a cell depends upon the old value, $\phi_{\rm old}$, the computed change in $\phi$, $\Delta \phi$, and the under-relaxation factor, $\alpha$, as follows:

 \phi = \phi_{\rm old} + \alpha \Delta \phi (25.4-22)

Under-Relaxation of Equations

The under-relaxation of equations, also known as implicit relaxation, is used in the pressure-based solver to stabilize the convergence behavior of the outer nonlinear iterations by introducing selective amounts of $\phi$ in the system of discretized equations. This is equivalent to the location-specific time step.

 \frac{a_p \phi}{\alpha} = \sum_{nb} a_{nb}\phi_{nb} + b + \frac{1-\alpha}{\alpha}a_{p}\phi_{\rm old} (25.4-23)

The CFL number is a solution parameter in the pressure-based coupled algorithm and can be written in terms of $\alpha$:

 \frac{1-\alpha}{\alpha} = \frac{1}{CFL} (25.4-24)

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