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MAIA bb96820c
Multiphysics at AIA
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The Ffowcs Williams and Hawkings (FWH) equation [WilliamHawkings1969] is obtained by an exact rearrangement of the Navier-Stokes equations, i.e, the conservation of mass, momentum, and energy, into an inhomogeneous wave equation featuring a non-linear right-hand side
\begin{align} \left( \fracpart{^2}{t^2} - c_s^2 \fracpart{^2}{x_i^2} \right) (H(s) \rhop) &= \fracpart {^2}{x_i x_j}(T_{ij} H(s)) - \fracpart{}{x_i} (F_i \delta(s)) + \fracpart{}{t} (Q \delta(s)) &&\\ T_{ij} &= \rho u_i u_j + P_{ij} - c_s^2 \rhop \delta_{ij} &&\text{(Quadrupole term)} \\ F_i &= (P_{ij} + \rho u_i (u_j - v_j)) \fracpart{s}{x_j} &&\text{(Dipole term)} \\ Q &= ( \rho_\infty v_i + \rho(u_i - v_i) ) \fracpart{s}{x_i} &&\text{(Monopole term)} \,, \end{align}
where \(\delta_{ij}\) is the Kronecker delta, \(\mm{P}=p\delta_{ij}\) is the compressive stress tensor, \(\rhop\) is the perturbed density, and \(H(\cdot)\) is the Heaviside step function. The function \(s\) is a support function, which takes zero value on a surface surrounding the volume of acoustic source terms \(V_a\), negative value in its inside, and positive values outside of this volume. From now on this surface is named FWH permeable surface. With \(H()\) being the Heaviside step function it follows that
\begin{equation*} H(s(\mv{x}, t)) = \begin{cases} 1, & \mv{x} \notin V_a \\ 0, & \mv{x} \in V_a \end{cases} \,. \end{equation*}
The FWH permeable surface \( s=0 \) is moving with the velocity \( \mv{v} \).
Following [Lockard2000], the FWH equation can be solved more efficiently in frequency domain than in time domain. Therefore, a uniform rectilinear motion of the surface \( s=0 \) with the speed \( \mv{U} \) needs to be assumed. Applying a Galilean transformation
\begin{equation*} \mv{y} = \mv{x} + \mv{U} t , \qquad \fracpart{}{x_i} = \fracpart{}{y_i} , \qquad \fracpart{}{t} = \fracpart{}{t'} + U_i \fracpart{}{y_i} \end{equation*}
and a Fourier transformation, the final governing equation consists of two surface integrals and one volume integral reading
\begin{aligned} H(s)c_s^2 \rhop(\mv{y_o}, \omega) = &- \oint_{s=0} \iu \omega Q(\mv{y_s}, \omega) G(\mv{y_o}; \mv{y_s}) dA \\ &- \oint_{s=0} F_i(\mv{y_s}, \omega) \fracpart{G(\mv{y_o}; \mv{y_s})}{y_{s,i}} dA \\ &- \int_{s>0} T_{ij}(\mv{y_s}, \omega) H(s) \fracpart{^2G(\mv{y_o}; \mv{y_s})}{y_{s,i} \partial y_{s,j}} \mv{dy_s} \,, \end{aligned}
with the angular frequency \(\omega\) and the Green's function \(G(\cdot;\cdot)\). Source and observer coordinates are given by \(\mv{y_s}\) and \(\mv{y_o}\), respectively. Here, observer refers to location of interests where the acoustic signal is to be investigated and source describes all locations contributing to the resulting signal.
As the term \( H(s(\mv{x}))=0 , \forall \mv{x} \in V_A \) the sum of the three integrals must zero inside of the volume of acoustic source terms per definition.
In it's integral form the FWH equation features two surface integrals which are to be solved on the chosen FWH permeable surface and one volume integral to be solved only in the volume that is not enclosed by the FWH permeable surface. Thus, if all significant contributions of acoustic quadrupole sources represented by the Lighthill stress tensor \( T_{ij} \) being part of the volume integral are contained within the enclosed volume it is a good and common approximation to neglect the calculation of the volume integral.
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