- Generated on Sun Jun 16 2024 23:45:16 for MAIA by
1.9.5
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MAIA bb96820c
Multiphysics at AIA
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In the ACA solver the variables are made non-dimensional with a infinity state, see Tab.1. The non-dimensionalization of the relevant variables is given in Tab.2.
| Free-stream speed of sound | \( c_\infty \) |
| Free-stream density | \( \rho_\infty \) |
| Reference length | \( L_{ref} = 1.0 \) (in STL file units) |
| Variable | Equation |
|---|---|
| Spatial coordinates | \( \mv{x}^* = \mv{x} \cdot \frac{1}{L} \) |
| Perturbed velocity | \( \mv{u'}^* = \mv{u'} \cdot \frac{1}{c_\infty} \) |
| Perturbed density | \( \rho'^* = \rho' \cdot \frac{1}{\rho_\infty} \) |
| Perturbed pressure | \( p'^* = p' \cdot \frac{1}{\rho_\infty c_\infty^2} \) |
| Time | \( t^* = t \cdot \frac{c_\infty}{L_{ref}} \) |
Solving the FWH equation numerically is computational more efficient in frequency domain [Lockard2000]. The Fourier transformation is a mathematical tool to perform this transformation and is defined in the following.
A signal in time domain \( q(t) \) is transformed into frequency domain \( Q(\omega) \) using the Fourier transformation \( \mathcal{F}(\cdot) \),
\begin{equation*} \mathcal{F}(q(t)) := \int_{-\infty}^{\infty} q(t) e^{-\iu\omega t} dt = Q(\omega) \,, \end{equation*}
where \( t \), \( f \), and \( \omega = 2\pi f \) are the time, frequency, and angular frequency, respectively.
The inverse Fourier transformation \( \mathcal{F^{-1}}(\cdot) \), i.e., the transformation from frequency into time domain, is defined as
\begin{equation*} \mathcal{F}^{-1}(Q(\omega)) := \frac{1}{2\pi}\int_{-\infty}^{\infty} Q(\omega) e^{\iu\omega t} d\omega = q(t) \,. \end{equation*}
The Fourier transformation is defined for functional signal over an infinite sampling duration. However, practical application provide discrete and finite samplings making the FT impractical.
The discretized Fourier transformation (DFT) is defined for a finite duration \( T \) with \( N \) equidistant samples spaced with \( \Delta t = \frac{T}{N} \). Defining the discrete frequency \( f_n \) and time \( t_k \)
\begin{align*} f_n &:= \frac{n}{N\dt} , \, \text{for} \quad n \in \mathbb{N} \land n\in \left[ -\frac{N}{2}, \frac{N}{2} \right] \,,\\ % t_k &:= k \Delta t , \, \text{for} \quad k \in \mathbb{N} \land n\in [0, N-1] \,,\\ \end{align*}
and introducing the abbrevations
\begin{align*} Q_n &:= Q(f_n) \\ q_k &:= q(t_k) \end{align*}
the DFT reads
\begin{equation*} Q_n = \sum_{k_0}^{N-1} q_k e^{\iu 2\pi f_n t_k} \dt = \dt \sum_{k_0}^{N-1} q_k e^{\iu 2\pi \frac{k~n}{N}} \,. \end{equation*}
The inverse DFT is
\begin{equation*} q_k = \frac{1}{N} \sum_{n=0}^{N-1} Q_n e^{-\iu 2\pi \frac{k~n}{N}} \end{equation*}
The implementation of a DFT is straightforward but its computation is cumbersome for bigger problems. Hence, for number of samples corresponding to a power of two ( \( N=2^i \,, \text{for} \, i\ \in \mathbb{N} \) ) the Fast Fourier Transformation (FFT) algorithm [Press1992] is a more efficient alternative.
While in the FT an infinite signal length is required the DFT allows for a finite but periodic signal as input. If the provided sample is not consisting of multiple oscillation periods distortions are introduced - often called spectral leakage. This disturbance can be reduced by multiplying the signal with window function, which zeros the signal towards its start and end. While a rectangle window describes the raw signal the following functions are common
\begin{align*} \text{0. Rectangle : } w_k &= 1 \\ \text{1. Hanning : } w_k &= \frac{1}{2} \left( 1 - \cos \left( \frac{2\pi k}{N} \right) \right) \\ \text{2. Hamming : } w_k &= 0.54 - 0.45 \cos \left( \frac{2\pi k}{N} \right) \\ \text{3. Modified Hanning: } w_k &= \begin{cases} 1 \,, &\text{for} \, k \in \left[ \frac{N}{8}, \frac{7N}{8} \right] \\ \frac{1}{2} \left( 1 - \cos \left( \frac{8\pi k}{N} \right) \right) \,, &\text{else} \end{cases} \end{align*}
Applying the window function, i.e., multiplying it with the signal, reduces the energy contained in the signal. Energy conservation is ensured by scaling the data after the Fourier transformation by the factor \( 1/E_T \)
\begin{equation*} E_T = \sqrt{ \frac{1}{N} \sum_{k=0}^{N-1} |w_k|^2} \,. \end{equation*}
1.9.5