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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The momentum equation without body forces | |||
{{NumEqn|<math> | |||
\rho\frac{D\mathbf{v}}{Dt}=-\nabla p | |||
</math>}} | |||
Expanding the substantial derivative | |||
{{NumEqn|<math> | |||
\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p | |||
</math>}} | |||
The first and second law of thermodynamics gives | |||
{{NumEqn|<math> | |||
T\nabla s =\nabla h-\frac{\nabla p}{\rho} | |||
</math>}} | |||
Insert <math>\nabla p</math> from the momentum equation | |||
{{NumEqn|<math> | |||
T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v} | |||
</math>}} | |||
Definition of total enthalpy (<math>h_o</math>) | |||
{{NumEqn|<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) | |||
</math>}} | |||
The last term can be rewritten as | |||
\ | {{NumEqn|<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} | |||
</math>}} | |||
which gives | |||
{{NumEqn|<math> | |||
\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v} | |||
</math>}} | |||
\ | Insert <math>\nabla h</math> in the entropy equation gives | ||
{{NumEqn|<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}} | |||
</math>}} | |||
{{NumEqn|<math> | |||
T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t} | |||
</math>}} | |||
