Source: Schlottke-Lakemper2019
The acoustic perturbation equations (APE) are derived from the conservation equations and modified to retain only acoustic modes (EwertSchroeder03). The APE-4 system can be written in non-dimensional form as
\begin{align} \frac{\partial \mv{u}'}{\partial t} + \mv{\nabla} \left( \bar{\mv{u}} \cdot \mv{u}' + \frac{p'}{\bar{\rho}} \right) &= \mv{q}_m, \\ \frac{\partial p'}{\partial t} + \bar{c}^2 \mv{\nabla} \cdot \left( \bar{\rho}\mv{u}' + \bar{\mv{u}} \frac{p'}{\bar{c}^2}\right) &= 0. \end{align}
The unknowns of the APE are acoustic velocity and pressure fluctuations, which are denoted by prime (\cdot)'. They are defined by \phi' := \phi - \bar{\phi}, where the bar \overline{(\cdot)} indicates mean quantities that have to be determined prior to the CAA simulation by time-averaging the flow solution. In the momentum equations, only the contribution of the Lamb vector source is usually considered for problems with noise generation dominated by vorticity, which is given by \begin{equation} \mv{q}_m = - (\mv{\omega} \times \mv{u})',\label{eqn:ape_q_m}\ \end{equation}
\begin{align} \frac{\partial \mv{u}'}{\partial t} + \mv{\nabla} \left( \bar{\mv{u}} \cdot \mv{u}' + \frac{p'}{\bar{\rho}} \right) &= \mv{q}_m, \\ \frac{\partial p'}{\partial t} + \mv{\nabla} \cdot \left( \bar{c}^2\bar{\rho}\mv{u}' + \bar{\mv{u}} p'\right) - \underbrace{\left(\bar{\rho}\mv{u}' + \bar{\mv{u}} \frac{p'}{\bar{c}^2} \right) \cdot \nabla \bar{c}^2}_{s_\text{cons}} &= 0, \end{align}
where s_\text{cons} is an additional term introduced by reformulating the APE-4 system.
1.9.5