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Time Derivative (Simplified PCWE Source Term)

For low flow velocities convective effects can be neglected for the PCWE (see Aeroacoustic Simulations), and the resulting source term simplifies to just the time derivative of the incompressible pressure \partial p^{\mathrm{ic}}/\partial t [^1]. This is the reason why the time derivative filter is placed in the section Aeroacoustic Source Terms. Of course, the filter can be applied to a different quanitity, of which a time derivative is required, as well.

The time derivative of the desired quantitiy \dot{q}(t) is computed by a smooth noise-robust differentiator, which supresses high frequencies and is precise on low frequencies, according to Holoborodko. For an efficient and robust calculation the order of the differentiator is set to N=5 and the time derivative is calculated by \begin{equation} \dot{q}(t)=\frac{2(q_1-q_{-1})+q_2-q_{-2}}{8 \Delta t}, \end{equation} where the index of q defines the time step relative to the time step of which the derivative is calculated and \Delta t is the time step size. The computation only requires the definition of the input quantity (inputQuantity-tag) and the desired name of the output quantity (outputQuantity-tag) as indicated below.

<timeDeriv1 id="TimeDerivative" inputFilterIds="input">
        <inputQuantity resultName="fluidMechPressure"/>
        <outputQuantity resultName="acouRhsLoadP"/>


Please provide an acknowledgement at the end of your publication using this software part for simulations

The computational results presented have been achieved [in part] using the software openCFS [Time Derivative].


[^1]: Stefan Schoder and Manfred Kaltenbacher. Hybrid aeroacoustic computations: state of art and new achievements. Journal of Theoretical and Computational Acoustics, 274:1950020, 2019.