Crocco's equation: Difference between revisions

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Revision as of 06:42, 17 March 2026

The momentum equation without body forces

$ρ \frac{D 𝐯}{D t} = - \nabla p$

Expanding the substantial derivative

$ρ \frac{\partial 𝐯}{\partial t} + ρ 𝐯 \cdot \nabla 𝐯 = - \nabla p$

The first and second law of thermodynamics gives

$T \nabla s = \nabla h - \frac{\nabla p}{ρ}$

Insert $\nabla p$ from the momentum equation

$T \nabla s = \nabla h + \frac{\partial 𝐯}{\partial t} + 𝐯 \cdot \nabla 𝐯$

Definition of total enthalpy ( $h_{o}$ )

$h_{o} = h + \frac{1}{2} 𝐯 \cdot 𝐯 \Rightarrow \nabla h = \nabla h_{o} - \nabla (\frac{1}{2} 𝐯 \cdot 𝐯)$

The last term can be rewritten as

$\nabla (\frac{1}{2} 𝐯 \cdot 𝐯) = 𝐯 \times (\nabla \times 𝐯) + 𝐯 \cdot \nabla 𝐯$

which gives

$\nabla h = \nabla h_{o} - 𝐯 \times (\nabla \times 𝐯) - 𝐯 \cdot \nabla 𝐯$

Insert $\nabla h$ in the entropy equation gives

$T \nabla s = \nabla h_{o} - 𝐯 \times (\nabla \times 𝐯) - 𝐯 \cdot \nabla 𝐯 + \frac{\partial 𝐯}{\partial t} + 𝐯 \cdot \nabla 𝐯$

$T \nabla s = \nabla h_{o} - 𝐯 \times (\nabla \times 𝐯) + \frac{\partial 𝐯}{\partial t}$

@@ Line 4: / Line 4: @@
 __TOC__
-\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>
-\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\
-\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\

Crocco's equation: Difference between revisions

Revision as of 06:42, 17 March 2026

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