Linear methods from quantum mechanics
Mathematical solution of indirect noise
From Magri, L., On indirect noise in multicomponent nozzle flows, Journal of Fluid Mechanics (2017), vol. 828, R2 (Rapids).
First, the sound produced by a multi-component gas inhomogeneity being accelerated in a nozzle is governed by
\begin{align} \label{eq:systAA}
2\pi\mathrm{i}He \mathbf{A}(\eta)\hat{{\mathcal{I}}}= \frac{d \hat{{\mathcal{I}}}}{d\eta}
\end{align}
where \[\bar{M}\not=1\] and \[\mathbf{A}=\mathbf{E}^{-1}\] reads
\begin{align}\label{eq:systAfA}
&\mathbf{A}(\eta)=\nonumber\\
&-\frac{1}{\tilde{u}}\left[
\small{\small{\begin{array}{cccc}
\frac{\bar{M}^2}{\bar{M}^2-1} & -\frac{\beta}{(\bar{\gamma}-1)(\bar{M}^2-1)} & \frac{\bar{\gamma}}{(\bar{\gamma}-1)(\bar{M}^2-1)} & \frac{\left(\bar{\Psi}+ \bar{\aleph}\right)\bar{\gamma}-\bar{\aleph}}{(\bar{\gamma}-1)(\bar{M}^2-1)} \\
-\frac{(\bar{\gamma}-1)\bar{M}^2}{(\bar{M}^2-1)\beta} & \frac{\bar{M}^2}{\bar{M}^2-1} & -\frac{1+(\bar{\gamma}-1)\bar{M}^2}{(\bar{M}^2-1)\beta} & -\frac{\left(\bar{\Psi}+ \bar{\aleph}\right)\bar{M}^2\left(\bar{\gamma}-1\right) + \bar{\Psi}}{(\bar{M}^2-1)\beta}\\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}}}
\right]
\end{align}
This matrix equation represents a set of four linear ordinary differential equations with spatially dependent coefficients.
If \[\left[\mathbf{A}(\eta_1),\mathbf{A}(\eta_2)\right]=\mathbf{A}(\eta_1)\mathbf{A}(\eta_2) - \mathbf{A}(\eta_2)\mathbf{A}(\eta_1)=0\] the formal solution reads
\[\hat{{\mathcal{I}}}=\exp\left(2\pi\mathrm{i}He\int_{\eta_a}^{\eta}\mathbf{A}(\eta')d\eta'\right)\hat{{\mathcal{I}}}_a\]
where \[\exp(\cdot)\] is the matrix exponential.
However, if the commutator is not zero, for example in the acoustic flow of this paper, the matrix-exponential solution no longer holds, because \[\exp\left(\mathbf{A}(\eta_1)\right)\exp\left(\mathbf{A}(\eta_2)\right)\not=\exp\left(\mathbf{A}(\eta_1)+\mathbf{A}(\eta_2)\right)\]
A solution for this case is derived by asymptotic expansion.
The governing equation is recast in integral form as
\begin{align}\label{eq:forsolA}
\hat{{\mathcal{I}}}(\eta) = \hat{{\mathcal{I}}}_a+ 2\pi\mathrm{i}He\int^{\eta}_{\eta_a}\mathbf{A}(\eta')\hat{{\mathcal{I}}}(\eta') d\eta'
\end{align}
which enables an explicit expression for the solution by recursion.
By recognizing the Helmholtz number as the perturbation parameter, the solution is expanded as
\begin{align}\label{eq:expansion}
& \hat{{\mathcal{I}}} = \hat{{\mathcal{I}}}_{a} + \sum_{n=1}^{\infty}He^n\hat{{\mathcal{I}}}_{n}
\end{align}
The asymptotic decomposition is substituted into the governing equation to yield
\begin{align}\label{eq:recursion_U}
&\small{\hat{{\mathcal{I}}}(\eta) = }\nonumber\\
& \small{\underbrace{\Bigg[\mathbf{1} + 2\pi \mathrm{i}He\int^{\eta}_{\eta_a}d\eta^{(1)}\mathbf{A}\left(\eta^{(1)}\right)
+\ldots+ ( 2\pi \mathrm{i}He)^n\int^{\eta}_{\eta_a}d\eta^{(1)}\ldots\int_{\eta_a}^{\eta^{(n-1)}} d\eta^{(n)}\mathbf{A}\left(\eta^{(1)}\right)\dots\mathbf{A}\left(\eta^{(n)}\right)\Bigg]}_{\textrm{Acoustic propagator}, \;\;\; \mathbf{U}=\mathbf{1} + \sum_{n=1}^{\infty}\left(2\pi\mathrm{i}He\right)^n\mathbf{P}_n}\hat{{\mathcal{I}}}_a}
\end{align}
where \[\eta_a<\eta^{(n)}<\ldots<\eta^{(1)}<\eta\] and \[\mathbf{1}\] is the identity operator. The integral operators are path-ordered, which means that the operator closer to the nozzle inlet \[\eta=\eta_a\] is always on the right of the operator acting at a farther location.
The solution contains the Neumann series of the acoustic propagator \[\mathbf{U}=\mathbf{1} + \sum_{n=1}^{\infty}\left(2\pi\mathrm{i}He\right)^n\mathbf{P}_n\] defined as the map such that \[\hat{{\mathcal{I}}}(\eta) = \mathbf{U}(\eta)\hat{{\mathcal{I}}}_a\]
Although asymptotic, the solution is absolutely convergent in a finite spatial domain and when \[\mathbf{A}\] is bounded, which is the case here.
This can be shown by defining the path-ordering operator
\[\mathcal{P}\left(\mathbf{P}(\eta_1),\mathbf{P}(\eta_2)\right) = \mathbf{P}(\eta_1)\mathbf{P}(\eta_2)\] if \[\eta_1>\eta_2\] and
\[\mathcal{P}\left(\mathbf{P}(\eta_1),\mathbf{P}(\eta_2)\right)=\mathbf{P}(\eta_2)\mathbf{P}(\eta_1)\] if \[\eta_2>\eta_1\]
After some algebra, it can be shown that
\begin{align}
\hat{{\mathcal{I}}} = \mathcal{P}\left(2\pi \mathrm{i}He\int^{\eta}_{\eta_a}\exp\left(\mathbf{A}\left(\eta'\right)\right)d\eta'\right)\hat{{\mathcal{I}}}_a
\end{align}
which means that the series
\begin{align}
\lVert \hat{{\mathcal{I}}} \rVert < \exp\left(\int_{{\eta_a}}^{\eta}d\eta'\lVert\mathbf{A}(\eta')\rVert\right)\lVert\hat{{\mathcal{I}}}_a\rVert<\infty
\end{align}
is absolutely convergent.
Once the acoustic propagator is calculated, (i) the solution can be calculated without iterative shooting methods for any boundary conditions; and (ii) the effect of the Helmholtz number can be directly isolated at each order.
In time-dependent perturbations of quantum systems, the aeroacoustic governing equation has an analogy to the time-dependent evolution of the perturbed Schroedinger equation with the Dirac picture, the acoustic solution has an analogy to the Dyson series, while the integrands have analogies to the Feynman path integrals. With the solution we derived, the nozzle transfer functions can be practically calculated.
First, the sound produced by a multi-component gas inhomogeneity being accelerated in a nozzle is governed by
\begin{align} \label{eq:systAA}
2\pi\mathrm{i}He \mathbf{A}(\eta)\hat{{\mathcal{I}}}= \frac{d \hat{{\mathcal{I}}}}{d\eta}
\end{align}
where \[\bar{M}\not=1\] and \[\mathbf{A}=\mathbf{E}^{-1}\] reads
\begin{align}\label{eq:systAfA}
&\mathbf{A}(\eta)=\nonumber\\
&-\frac{1}{\tilde{u}}\left[
\small{\small{\begin{array}{cccc}
\frac{\bar{M}^2}{\bar{M}^2-1} & -\frac{\beta}{(\bar{\gamma}-1)(\bar{M}^2-1)} & \frac{\bar{\gamma}}{(\bar{\gamma}-1)(\bar{M}^2-1)} & \frac{\left(\bar{\Psi}+ \bar{\aleph}\right)\bar{\gamma}-\bar{\aleph}}{(\bar{\gamma}-1)(\bar{M}^2-1)} \\
-\frac{(\bar{\gamma}-1)\bar{M}^2}{(\bar{M}^2-1)\beta} & \frac{\bar{M}^2}{\bar{M}^2-1} & -\frac{1+(\bar{\gamma}-1)\bar{M}^2}{(\bar{M}^2-1)\beta} & -\frac{\left(\bar{\Psi}+ \bar{\aleph}\right)\bar{M}^2\left(\bar{\gamma}-1\right) + \bar{\Psi}}{(\bar{M}^2-1)\beta}\\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}}}
\right]
\end{align}
This matrix equation represents a set of four linear ordinary differential equations with spatially dependent coefficients.
If \[\left[\mathbf{A}(\eta_1),\mathbf{A}(\eta_2)\right]=\mathbf{A}(\eta_1)\mathbf{A}(\eta_2) - \mathbf{A}(\eta_2)\mathbf{A}(\eta_1)=0\] the formal solution reads
\[\hat{{\mathcal{I}}}=\exp\left(2\pi\mathrm{i}He\int_{\eta_a}^{\eta}\mathbf{A}(\eta')d\eta'\right)\hat{{\mathcal{I}}}_a\]
where \[\exp(\cdot)\] is the matrix exponential.
However, if the commutator is not zero, for example in the acoustic flow of this paper, the matrix-exponential solution no longer holds, because \[\exp\left(\mathbf{A}(\eta_1)\right)\exp\left(\mathbf{A}(\eta_2)\right)\not=\exp\left(\mathbf{A}(\eta_1)+\mathbf{A}(\eta_2)\right)\]
A solution for this case is derived by asymptotic expansion.
The governing equation is recast in integral form as
\begin{align}\label{eq:forsolA}
\hat{{\mathcal{I}}}(\eta) = \hat{{\mathcal{I}}}_a+ 2\pi\mathrm{i}He\int^{\eta}_{\eta_a}\mathbf{A}(\eta')\hat{{\mathcal{I}}}(\eta') d\eta'
\end{align}
which enables an explicit expression for the solution by recursion.
By recognizing the Helmholtz number as the perturbation parameter, the solution is expanded as
\begin{align}\label{eq:expansion}
& \hat{{\mathcal{I}}} = \hat{{\mathcal{I}}}_{a} + \sum_{n=1}^{\infty}He^n\hat{{\mathcal{I}}}_{n}
\end{align}
The asymptotic decomposition is substituted into the governing equation to yield
\begin{align}\label{eq:recursion_U}
&\small{\hat{{\mathcal{I}}}(\eta) = }\nonumber\\
& \small{\underbrace{\Bigg[\mathbf{1} + 2\pi \mathrm{i}He\int^{\eta}_{\eta_a}d\eta^{(1)}\mathbf{A}\left(\eta^{(1)}\right)
+\ldots+ ( 2\pi \mathrm{i}He)^n\int^{\eta}_{\eta_a}d\eta^{(1)}\ldots\int_{\eta_a}^{\eta^{(n-1)}} d\eta^{(n)}\mathbf{A}\left(\eta^{(1)}\right)\dots\mathbf{A}\left(\eta^{(n)}\right)\Bigg]}_{\textrm{Acoustic propagator}, \;\;\; \mathbf{U}=\mathbf{1} + \sum_{n=1}^{\infty}\left(2\pi\mathrm{i}He\right)^n\mathbf{P}_n}\hat{{\mathcal{I}}}_a}
\end{align}
where \[\eta_a<\eta^{(n)}<\ldots<\eta^{(1)}<\eta\] and \[\mathbf{1}\] is the identity operator. The integral operators are path-ordered, which means that the operator closer to the nozzle inlet \[\eta=\eta_a\] is always on the right of the operator acting at a farther location.
The solution contains the Neumann series of the acoustic propagator \[\mathbf{U}=\mathbf{1} + \sum_{n=1}^{\infty}\left(2\pi\mathrm{i}He\right)^n\mathbf{P}_n\] defined as the map such that \[\hat{{\mathcal{I}}}(\eta) = \mathbf{U}(\eta)\hat{{\mathcal{I}}}_a\]
Although asymptotic, the solution is absolutely convergent in a finite spatial domain and when \[\mathbf{A}\] is bounded, which is the case here.
This can be shown by defining the path-ordering operator
\[\mathcal{P}\left(\mathbf{P}(\eta_1),\mathbf{P}(\eta_2)\right) = \mathbf{P}(\eta_1)\mathbf{P}(\eta_2)\] if \[\eta_1>\eta_2\] and
\[\mathcal{P}\left(\mathbf{P}(\eta_1),\mathbf{P}(\eta_2)\right)=\mathbf{P}(\eta_2)\mathbf{P}(\eta_1)\] if \[\eta_2>\eta_1\]
After some algebra, it can be shown that
\begin{align}
\hat{{\mathcal{I}}} = \mathcal{P}\left(2\pi \mathrm{i}He\int^{\eta}_{\eta_a}\exp\left(\mathbf{A}\left(\eta'\right)\right)d\eta'\right)\hat{{\mathcal{I}}}_a
\end{align}
which means that the series
\begin{align}
\lVert \hat{{\mathcal{I}}} \rVert < \exp\left(\int_{{\eta_a}}^{\eta}d\eta'\lVert\mathbf{A}(\eta')\rVert\right)\lVert\hat{{\mathcal{I}}}_a\rVert<\infty
\end{align}
is absolutely convergent.
Once the acoustic propagator is calculated, (i) the solution can be calculated without iterative shooting methods for any boundary conditions; and (ii) the effect of the Helmholtz number can be directly isolated at each order.
In time-dependent perturbations of quantum systems, the aeroacoustic governing equation has an analogy to the time-dependent evolution of the perturbed Schroedinger equation with the Dirac picture, the acoustic solution has an analogy to the Dyson series, while the integrands have analogies to the Feynman path integrals. With the solution we derived, the nozzle transfer functions can be practically calculated.