10.2.2: Compressible Flow Stream Function
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The stream function can be defined also for the compressible flow substances and steady state. The continuity equation is used as the base for the derivations. The continuity equation for compressible substance is
\[
\label{if:eq:continutyRho}
\dfrac{\partial \rho\, \pmb{U}_x}{ dx} + \dfrac{\partial \rho\, \pmb{U}_y}{ dy} = 0 \tag{59}
\]
To absorb the density, dimensionless density is inserted into the definition of the stream function as
\[
\label{if:eq:rhoStreamFunY}
\dfrac{\partial \psi }{ dy} = \dfrac{\rho\, U_x}{\rho_0} \tag{60}
\]
and
\[
\label{if:eq:rhoStreamFunX}
\dfrac{\partial \psi }{ dx} = \dfrac{\rho\, U_y}{\rho_0} \tag{61}
\]
Where \(\rho_0\) is the density at a location or a reference density. Note that the new stream function is not identical to the previous definition and they cannot be combined. The stream function, as it was shown earlier, describes (constant) stream lines. Using the same argument in which equation (50) and equation (??) were developed leads to equation (53) and there is no difference between compressible flow and incompressible flow case. Substituting equations (60) and (61) into equation (53) yields
\[
\label{if:eq:streamUcompressible}
\left( \dfrac{\partial \psi}{\partial y} \,dy +
\dfrac{\partial \psi}{\partial x} \,dx \right)\, \dfrac{\rho_0}{\rho} =
\dfrac{\rho_0}{\rho} \, d\psi \tag{62}
\]
Equation suggests that the stream function should be redefined so that similar expressions to incompressible flow can be developed for the compressible flow as
\[
\label{if:eq:compressibleFlowStreamFun}
d\psi = \dfrac{\rho_0}{\rho} \, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d\ell \tag{63}
\]
With the new definition, the flow crossing the line \(1\) to \(2\), utilizing the new definition of (63) is
\[
\label{if:eq:mDOTcompressibleFlow}
\dot{m} = \int_1^2 \rho\, \pmb{U} \boldsymbol{\cdot} \widehat{s} \, d'\ell =
\rho_0 \int_1^2 d\psi = \rho_0 \left( \psi_2 \psi_1 \right) \tag{64}
\]