Crocco's equation - Revision history

Nian at 05:21, 1 April 2026

2026-04-01T05:21:16Z

← Older revision		Revision as of 05:21, 1 April 2026
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	The momentum equation without body forces		The momentum equation without body forces

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\rho\frac{D\mathbf{v}}{Dt}=-\nabla p		\rho\frac{D\mathbf{v}}{Dt}=-\nabla p
	</math>		</math>}}

	Expanding the substantial derivative		Expanding the substantial derivative

	<math ~~display="block"~~>		{{NumEqn\|<math>
	\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p		\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p
	</math>		</math>}}

	The first and second law of thermodynamics gives		The first and second law of thermodynamics gives

	<math ~~display="block"~~>		{{NumEqn\|<math>
	T\nabla s =\nabla h-\frac{\nabla p}{\rho}		T\nabla s =\nabla h-\frac{\nabla p}{\rho}
	</math>		</math>}}

	Insert <math>\nabla p</math> from the momentum equation		Insert <math>\nabla p</math> from the momentum equation

	<math ~~display="block"~~>		{{NumEqn\|<math>
	T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}		T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}
	</math>		</math>}}

	Definition of total enthalpy (<math>h_o</math>)		Definition of total enthalpy (<math>h_o</math>)

	<math ~~display="block"~~>		{{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)		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>		</math>}}

	The last term can be rewritten as		The last term can be rewritten as

	<math ~~display="block"~~>		{{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}		\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>		</math>}}

	which gives		which gives

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

	Insert <math>\nabla h</math> in the entropy equation gives		Insert <math>\nabla h</math> in the entropy equation gives

	<math ~~display="block"~~>		{{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}}		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>		</math>}}

	<math ~~display="block"~~>		{{NumEqn\|<math>
	T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}		T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}
	</math>		</math>}}

Nian at 19:29, 20 March 2026

2026-03-20T19:29:34Z

← Older revision		Revision as of 19:29, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Governing equations]]
			<noinclude>
	[[Category:Compressible flow:Topic]]		[[Category:Compressible flow:Topic]]
	~~[[Category:Governing equations]]~~		</noinclude>

	__TOC__		__TOC__

Nian at 18:57, 20 March 2026

2026-03-20T18:57:07Z

← Older revision		Revision as of 18:57, 20 March 2026
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	[[Category:Topic]]		[[Category:Compressible flow:Topic]]
	[[Category:Governing equations]]		[[Category:Governing equations]]

Nian at 15:47, 20 March 2026

2026-03-20T15:47:57Z

← Older revision		Revision as of 15:47, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:Governing equations]]		[[Category:Governing equations]]

Nian at 06:42, 17 March 2026

2026-03-17T06:42:13Z

← Older revision		Revision as of 06:42, 17 March 2026
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	__TOC__		__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>

	\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})~~-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}~~+\frac{\partial \mathbf{v}}{\partial t}~~+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\~~

	~~\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\~~

Nian: 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..."

2026-03-16T13:41:55Z

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..."

New page

[[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 $\nabla p$ from the momentum equation\\

\[T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent Definition of total enthalpy ($h_o$)\\

\[h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)\]\\

%\newpage

\noindent The last term can be rewritten as\\

\[\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent which gives\\

\[\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent Insert $\nabla h$ in the entropy equation gives\\

\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\

\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\