jet
===

class description
-----------------

.. automodule:: jet
	:members:
	:undoc-members:
	:show-inheritance:
	:exclude-members:

theory
------

Jet mixing noise
""""""""""""""""

Jet mixing noise accounts for the noise generated by turbulent jet fluid exiting a nozzle creating a shear layer with the surrounding fluid. The SAE ARP876 method is used to estimate mean-square acoustic pressure of the the jet mixing noise:

.. math::

    <p^2(r_s^*)>^* = 10\log_{10}\left[\frac{\Pi^*_{jet}A_{jet}^*}{4\pi (r_s^*)^2p_{\textrm{ref}}^2}  \frac{D(\theta,V_{jet}^*) F(S_c, \theta,V_{jet}^*,T_{jet}^*)}{1-M_0 \cos(\theta-\delta)} \left(\frac{V_{jet}^*-M_0}{V_{jet}^*}\right)^{m(\theta)} \right]

where:

.. math::

     \Pi_{jet}^* = (6.67\cdot 10^{-5}) (\rho_{jet}^*)^{\omega(V_{jet}^*)} (V_{jet}^*)^8 P(V_{jet}^*) 

The density exponent, :math:`\omega`, accounts for the effect of density on noise in heated jets. The Strouhal number, :math:`S_c`, used in the spectral distribution function, :math:`F(S_c, \theta,V_{jet}^*,T_{jet}^*)`, is given by :math:`S_c = \frac{f^*d_{jet}^*}{\xi(V_{jet}^*)(V_{jet}^*-M_0)}`. The directivity function, :math:`D(\theta, V_{jet}^*)`, the spectral distribution, :math:`F(S_c,\theta,V_{jet}^*,T_{jet}^*)`, the density exponent, :math:`\omega(V_j^*)`, the power deviation factor :math:`P(V_{j}^*)`, the Strouhal correction factor :math:`\xi(V_{jet}^*)`, and the forward velocity index :math:`m(\theta)` are tabulated online.

Jet shock-cell noise
""""""""""""""""""""

Supersonic jets that are not perfectly expanded will create a shocks cell structure that interact with the turbulent jet flow. This interaction is the main driver of the shock-cell noise. The SAE ARP876 method is used to estimate jet-shock cell mean-square acoustic pressure for supersonic exhaust Mach numbers (:math:`M_{jet}>1`):

.. math::

    <p^2(r_s^*)>^* =  10\log_{10} \left[ \frac{1.92\cdot10^{-3} A_j^*}{4\pi(r_s^*)^2p_{\textrm{ref}}^2} \left[\frac{1+W(\sigma,\theta,V_j^*)}{1-M_0 \cos(\theta-\delta)}\right] \beta^\eta H(\sigma) \right]

The pressure ratio parameter, :math:`\beta = \sqrt{M_{jet}^2-1}`, indicates that shock noise will only be produced with a supersonic exhaust velocity. The frequency parameter, :math:`\sigma = 7.8\beta(1-M_0 \cos \theta) \sqrt{A_{jet}^*}f^*`, where :math:`f^* = f\sqrt{A_e}/c_0`.
The pressure ratio parameter, :math:`\eta`, is given by:

.. math::

	\eta = \begin{cases}
	    1  &  \textrm{if } T_{jet}^* < 1.1 \text{ and } \beta > 1 \\
	    2  &  \textrm{if } T_{jet}^* \leq 1.1 \text{ and } \beta > 1 \\
	    4  &  \textrm{if } \beta \leq 1
	\end{cases}

The shock-cell interference function, :math:`W(\sigma, \theta, V_{jet}^*)`, is given by:\footnote{This equation is an updated version of Eq. 6 on page 8.5-4 in Zorumski. To obtain the shock cell interference function graph on Figure 4 in Zorumski, one factor \textit{b} in the denominator should be omitted.}

.. math::

	\begin{split}
	    W = \frac{4}{N_s b} \sum_{k=1}^{N_s-1} \left[C(\sigma\right]^{k^2} \sum_{m=0}^{N_s - (k+1)} \frac{\sin (b\sigma q_{km}/2)}{\sigma q_{km}} \cos(\sigma q_{km}) \hspace{0.5cm} \textrm{where} \\ \hspace{0.5cm} q_{km} = \frac{1.7k}{V_{jet}^*} \left[ 1 - 0.06 \left( m + \frac{k+1}{2} \right) \right](1 + 0.7V_{jet}^* \cos \theta)
	\end{split}

The correlation coefficient, :math:`C(\sigma)`, and the group source strength spectrum, :math:`H(\sigma)`, are tabulated online.