Crocco's equation: Difference between revisions

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Created page with "Category:Compressible flow Category:Governing equations __TOC__ \section{Crocco's Equation} \noindent The momentum equation without body forces\\ \[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p\]\\ \noindent Expanding the substantial derivative\\ \[\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p\]\\ \noindent The first and second law of thermodynamics gives\\ \[T\nabla s =\nabla h-\frac{\nabla p}{\rho}\]\\ \noindent Insert..."
 
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\section{Crocco's Equation}
The momentum equation without body forces


\noindent The momentum equation without body forces\\
<math display="block">
\rho\frac{D\mathbf{v}}{Dt}=-\nabla p
</math>


\[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p\]\\
Expanding the substantial derivative


\noindent Expanding the substantial derivative\\
<math display="block">
\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p
</math>


\[\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p\]\\
The first and second law of thermodynamics gives


\noindent The first and second law of thermodynamics gives\\
<math display="block">
T\nabla s =\nabla h-\frac{\nabla p}{\rho}
</math>


\[T\nabla s =\nabla h-\frac{\nabla p}{\rho}\]\\
Insert <math>\nabla p</math> from the momentum equation


\noindent Insert $\nabla p$ from the momentum equation\\
<math display="block">
T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}
</math>


\[T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\]\\
Definition of total enthalpy (<math>h_o</math>)


\noindent Definition of total enthalpy ($h_o$)\\
<math display="block">
h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)
</math>


\[h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)\]\\
The last term can be rewritten as


%\newpage
<math display="block">
\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}
</math>


\noindent The last term can be rewritten as\\
which gives


<math display="block">
\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}
</math>


\[\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}\]\\
Insert <math>\nabla h</math> in the entropy equation gives


\noindent which gives\\
<math display="block">
T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}
</math>


\[\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}\]\\
<math display="block">
 
T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}
\noindent Insert $\nabla h$ in the entropy equation gives\\
</math>
 
\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\
 
\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\
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