15.1.1. Steady flow
Integral balance equations.
\[\begin{split}\begin{aligned}
& \dfrac{d}{dt} \int_{v_t} \rho + \oint_{\partial v_t} \rho \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = 0 \\
& \dfrac{d}{dt} \int_{v_t} \rho \mathbf{u} + \oint_{\partial v_t} \rho \mathbf{u} \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \\
& \dfrac{d}{dt} \int_{v_t} \rho e^t + \oint_{\partial v_t} \rho e^t \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} \cdot \mathbf{u} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \oint_{\partial v_t} \mathbf{q} \cdot \hat{\mathbf{n}} \\
\end{aligned}\end{split}\]
For a control volume \(v_t = V\) at rest, \(\mathbf{u}_b = \mathbf{0}\) and \(\mathbf{u}^{rel} = \mathbf{u}\). Neglecting the volume force \(\mathbf{g} = \mathbf{0}\), and integrating over an elementary control volume \(\Delta V\) cutting a streamtube at coordinates \(z\), \(z + \Delta z\), under steady conditions
\[\begin{split}\begin{aligned}
& \oint_{\partial \Delta V} \rho \mathbf{u} \cdot \hat{\mathbf{n}} = 0 \\
& \oint_{\partial \Delta V} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} = \oint_{\partial \Delta V} \mathbf{t}_{\mathbf{n}} \\
& \oint_{\partial \Delta V} \rho e^t \mathbf{u} \cdot \hat{\mathbf{n}} = \oint_{\partial \Delta V} \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u}
\end{aligned}\end{split}\]
Differential equations.
Assuming uniform properties on each section,
\[\begin{split}\begin{aligned}
& \dfrac{d}{dz} \left( \rho u A \right) = 0 \\
& \dfrac{d}{dz} \left( \rho u^2 A + p A \right) - p A' = 0 \\
& \dfrac{d}{dz} \left[ \rho \left( e^t + \frac{p}{\rho} \right) u A \right] = 0 \ .
\end{aligned}\end{split}\]
Thus, momentum equation can be recast as
\[0 = \left( \rho u^2 A \right)' + p' A \ .\]
\[\begin{split}\begin{aligned}
\mathbf{0}
& = \oint_{\partial \Delta V} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} - \oint_{\partial \Delta V} \mathbf{t}_{\mathbf{n}} = \\
& = \int_{A(z)} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} + p \hat{\mathbf{n}} + \int_{A(z+\Delta z)} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} + p \hat{\mathbf{n}} + \int_{\Delta A^{lat}} p \hat{\mathbf{n}} \ .
\end{aligned}\end{split}\]
For 2-dimensional flows,
\[\hat{\mathbf{z}} \cdot \int_{\Delta A^{lat}} p \hat{\mathbf{n}} = - \Delta z \int_{\ell^{lat}} p \hat{\mathbf{n}} \cdot \hat{\mathbf{z}} \simeq - \Delta z \, p \, \frac{A'(z) \Delta z}{\Delta z} = - p(z) A'(z) \Delta z \ . \]
For 3-dimensional flows,…
Using mass equation, \(( \rho u A )' = 0\), \(\rho u A =: \dot{m}\), momentum and total energy equations become
\[\begin{split}\begin{aligned}
& \rho u A u' + p' A = 0 \\
& \rho u A h^{t \, '} = 0 \ ,
\end{aligned}\end{split}\]
or, simplifying momentum equation for the area of the section of the stream-tube and total energy equation for a mass flux \(\dot{m} \ne 0\),
\[\begin{split}\begin{aligned}
& \rho u u' + p' = 0 \\
& h^{t \, '} = 0 \ .
\end{aligned}\end{split}\]
15.1.1.1. Mach dependence
Subsonic and supersonic flows show different behavior. If the differential equations hold (no shock, homoentropic flow, \(ds = 0\)), pressure derivative is proportional to density derivative through the square of the speed of sound,
\[d p = a^2(\rho, \overline{s}) d \rho \ .\]
\[\begin{split}\begin{aligned}
\frac{A'}{A}
& = \left( M^2 - 1 \right) \frac{u'}{u} \\
& = - \frac{ M^2 - 1 }{M^2} \frac{\rho'}{\rho} \\
\end{aligned}\end{split}\]
Thus,
\[\begin{split}\begin{aligned}
0 & = \rho u u' + a^2 \rho' && \text{(momentum)} \\
0 & = \frac{\rho'}{\rho} + \frac{u'}{u} + \frac{A'}{A} && \text{(mass)} \\
& = - \frac{u u'}{a^2} + \frac{u'}{u} + \frac{A'}{A} \\
\end{aligned}\end{split}\]
and thus
velocity
\[\left( M^2 - 1 \right) \frac{u'}{u} = \frac{A'}{A} \ .\]
Thus,
in a subsonic flow \(M < 1\), velocity increases as the section of the stream-tube decreases
in a supersonic flow \(M > 1\), velocity increases as the section of the stream-tube increases
density
\[\begin{split}\begin{aligned}
\frac{\rho'}{\rho} & = - \frac{u^2}{a^2} \frac{u'}{u} \\
- \frac{M^2 - 1}{M^2} \frac{\rho'}{\rho} & = - \frac{A'}{A} \\
\end{aligned}\end{split}\]
Thus,
in a subsonic flow \(M < 1\), density decreases as the section of the stream-tube decreases
in a supersonic flow \(M > 1\), density decreases as the section of the stream-tube increases
pressure
…
temperature
…
15.1.2. Unsteady flow
todo For a generic control volume \(v_t\)… It could be useful for mechanical/multi-field system dynamics…
Integral equations.
\[\begin{split}\begin{aligned}
& \dfrac{d}{dt} \int_{v_t} \rho + \oint_{\partial v_t} \rho \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = 0 \\
& \dfrac{d}{dt} \int_{v_t} \rho \mathbf{u} + \oint_{\partial v_t} \rho \mathbf{u} \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \\
& \dfrac{d}{dt} \int_{v_t} \rho e^t + \oint_{\partial v_t} \rho e^t \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} \cdot \mathbf{u} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \oint_{\partial v_t} \mathbf{q} \cdot \hat{\mathbf{n}} \\
\end{aligned}\end{split}\]
For a control volume \(v_t = V\) at rest, \(\mathbf{u}_b = \mathbf{0}\) and \(\mathbf{u}^{rel} = \mathbf{u}\). Neglecting the volume force \(\mathbf{g} = \mathbf{0}\), and integrating over an elementary control volume \(\Delta V\) cutting a streamtube at coordinates \(z\), \(z + \Delta z\), under steady conditions
\[\begin{split}\begin{aligned}
& \frac{d}{dt} \int_{\Delta V} \rho + \oint_{\partial \Delta V} \rho \mathbf{u} \cdot \hat{\mathbf{n}} = 0 \\
& \frac{d}{dt} \int_{\Delta V} \rho \mathbf{u} + \oint_{\partial \Delta V} \rho \mathbf{u} \mathbf{u} \cdot \hat{\mathbf{n}} = \oint_{\partial \Delta V} \mathbf{t}_{\mathbf{n}} \\
& \frac{d}{dt} \int_{\Delta V} \rho e^t + \oint_{\partial \Delta V} \rho e^t \mathbf{u} \cdot \hat{\mathbf{n}} = \oint_{\partial \Delta V} \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u}
\end{aligned}\end{split}\]
Differential equations.
Assuming uniform properties on each section,
\[\begin{split}\begin{aligned}
& \partial_t \left( \rho A \right) + \partial_z \left( \rho u A \right) = 0 \\
& \partial_t \left( \rho u A \right) + \partial_z \left[ \left( \rho u^2 + p \right) A \right] - p \partial_z A = 0 \\
& \partial_t \left( \rho e^t A \right) + \partial_z \left[ \rho \left( e^t + \frac{p}{\rho} \right) u A \right] = 0 \ .
\end{aligned}\end{split}\]
Assuming here rigid stream-tube with known section \(A(z)\), constant in time \(\partial_t A(z,t) = 0\), the differential equations can be recast as a set of PDE, with primary unknowns \((\rho A, m A, E^t A)\) whose
conservative form reads
\[\begin{split}
\partial_t \begin{bmatrix} \rho A \\ \rho u A \\ \rho e^t A \end{bmatrix} +
\partial_z \begin{bmatrix} \rho u A \\ \frac{( \rho u A )^2}{\rho A} + p A \\ \frac{ (\rho e^t A + p A) (\rho u A) }{ \rho A } \end{bmatrix}
= \begin{bmatrix} 0 \\ p \partial_z A \\ 0 \end{bmatrix}
\end{split}\]
conservative quasi-linear form reads
\[\begin{split}
\partial_t \begin{bmatrix} \rho A \\ \rho u A \\ \rho e^t A \end{bmatrix} +
\begin{bmatrix}
\cdot & 1 & \cdot \\
- \frac{(\rho u A)^2}{(\rho A)^2} + \partial_{\rho A} (pA) & 2 \frac{\rho u A}{\rho A} + \partial_{\rho u A} (pA) & \partial_{\rho e^t A} (pA) \\
- \frac{(\rho h^t A)(\rho u A)}{(\rho A)^2} + \partial_{\rho A} (pA) \frac{\rho u A}{\rho A} & \frac{\rho h^t A}{\rho A} + \partial_{\rho u A} ( \rho h^t A ) \frac{\rho u A}{\rho A} & \frac{\rho u A}{\rho A} \left( 1 + \partial_{\rho e^t} ( pA ) \right)
\end{bmatrix}
\partial_z \begin{bmatrix} \rho A \\ \rho u A \\ \rho e^t A \end{bmatrix}
= \begin{bmatrix} 0 \\ p \partial_z A \\ 0 \end{bmatrix}
\end{split}\]
convective form reads
\[\begin{split}\begin{cases}
A \partial_t \rho + A ( \rho \partial_z u + u \partial_z \rho ) = - \rho u \partial_z A \\
\rho A \partial_t u + \rho A u \partial_z u + A \partial_z p = 0 \\
\rho A \partial_t e^t + \rho A u \partial_z e^t + A \left( u \partial_z p + p \partial_z u \right) = - p u \partial_z A \ .
\end{cases}\end{split}\]
or
\[\begin{split}\begin{cases}
\partial_t \rho + \rho \partial_z u + u \partial_z \rho = - \rho u \frac{\partial_z A}{A} \\
\partial_t u + u \partial_z u + \frac{ \partial_z p }{\rho} = 0 \\
\partial_t e^t + u \partial_z e^t + \frac{u}{\rho} \partial_z p + \frac{p}{\rho} \partial_z u = - \frac{p u}{\rho} \frac{\partial_z A}{A} \ .
\end{cases}\end{split}\]
\[\begin{split}\begin{aligned}
\partial_t e
& = \partial_t e^t - \partial_t \frac{u^2}{2} = \\
& = -u \partial_z e^t - \frac{u}{\rho}\partial_z p - \frac{p}{\rho} \partial_z u - \frac{pu}{\rho} \frac{\partial_z A}{A} + u \partial_z \frac{u^2}{2} + \frac{u}{\rho} \partial_z p = \\
& = -u \partial_z e - \frac{p}{\rho} \partial_z u - \frac{pu}{\rho} \frac{\partial_z A}{A} \ ,
\end{aligned}\end{split}\]
i.e.
\[ \partial_t e + u \partial_z e = - \frac{p}{\rho} \partial_z u - \frac{pu}{\rho} \frac{\partial_z A}{A} \ .\]
\[\begin{split}\begin{aligned}
T D_t s
& = D_t e - \frac{p}{\rho^2} D_t \rho = \\
& = - \frac{p}{\rho} \partial_z u - \frac{pu}{\rho} \frac{\partial_z A}{A} - \frac{p}{\rho^2} \left( - \rho \partial_z u - \rho u \frac{\partial_z A}{A} \right) = \\
& = 0 \ ,
\end{aligned}\end{split}\]
or, as \(T > 0\),
\[\begin{split}\begin{aligned}
0
& = D_t s = \\
& = \partial_t s + u \partial_z s \ .
\end{aligned}\end{split}\]
todo Discuss homoentropic flow, if no shock occurs
Quasi linear form with mass, momentum and entropy equation and \((\rho, u, s)\) primary variables reads
\[\begin{split}
\partial_t \begin{bmatrix} \rho \\ u \\ s \end{bmatrix} +
\begin{bmatrix}
u & \rho & \cdot \\
\frac{1}{\rho}\left( \partial_\rho p \right)_s & u & \frac{1}{\rho}\left( \partial_s p \right)_\rho \\
\cdot & \cdot & u
\end{bmatrix}
\partial_z \begin{bmatrix} \rho \\ u \\ s \end{bmatrix} =
\begin{bmatrix} - \rho u \frac{\partial_z A}{A} \\ 0 \\ 0 \end{bmatrix} \ .
\end{split}\]
The derivative \((\partial_\rho p)_s = a^2\) is the square of the speed of sound, as its meaning becomes clear in the spectrum of the convection matrix.
These equations looks like 1-dimensional Euler equations for compressible ideal flows, with the exception that a non-uniform section of the stream-tube acts as a source term in mass equation.
Spectrum - eigenvalues.
\[0 = |\mathbf{A}(\mathbf{u}) - s \mathbf{I}| = ( u - s )^3 - ( u - s ) a^2 \ ,\]
i.e.
\[s_{1,3} = u \mp a \quad , \quad s_2 = u \ .\]
Right eigenvectors
\[\begin{split}
\mathbf{R} =
\begin{bmatrix}
\rho & \rho & \rho \\
-a & \cdot & a \\
\cdot & - \frac{\rho a^2}{(\partial_s p)_{\rho}} & \cdot
\end{bmatrix}
\end{split}\]
Left eigenvectors
\[\begin{split}
\mathbf{L} =
\begin{bmatrix}
\frac{1}{2 \rho} & - \frac{1}{2 a} & \frac{(\partial_s p)_\rho}{2 \rho a^2} \\
\cdot & \cdot & -\frac{(\partial_s p)_\rho}{ \rho a^2} \\
\frac{1}{2 \rho} & \frac{1}{2 a} & \frac{(\partial_s p)_\rho}{2 \rho a^2} \\
\end{bmatrix}
\end{split}\]
