Crocco's equation: Difference between revisions

Latest revision as of 05:21, 1 April 2026

The momentum equation without body forces

ρ \frac{D 𝐯}{D t} = - \nabla p

(Eq. 2.86)

Expanding the substantial derivative

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

(Eq. 2.87)

The first and second law of thermodynamics gives

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

(Eq. 2.88)

Insert $\nabla p$ from the momentum equation

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

(Eq. 2.89)

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)

(Eq. 2.90)

The last term can be rewritten as

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

(Eq. 2.91)

which gives

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

(Eq. 2.92)

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

(Eq. 2.93)

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

(Eq. 2.94)

@@ Line 1: / Line 1: @@
-[[Category:Compressible flow]]
+<!--
-[[Category:Governing equations]]
+-->[[Category:Compressible flow]]<!--
+-->[[Category:Governing equations]]<!--
+--><noinclude><!--
+-->[[Category:Compressible flow:Topic]]<!--
+--></noinclude><!--
-__TOC__
+--><nomobile><!--
+-->__TOC__<!--
+--></nomobile><!--
-\section{Crocco's Equation}
+--><noinclude><!--
+-->{{#vardefine:secno|2}}<!--
+-->{{#vardefine:eqno|85}}<!--
+--></noinclude><!--
-\noindent The momentum equation without body forces\\
+-->
+The momentum equation without body forces
-\[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p\]\\
+{{NumEqn|<math>
+\rho\frac{D\mathbf{v}}{Dt}=-\nabla p
+</math>}}
-\noindent Expanding the substantial derivative\\
+Expanding the substantial derivative
-\[\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p\]\\
+{{NumEqn|<math>
+\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p
+</math>}}
-\noindent The first and second law of thermodynamics gives\\
+The first and second law of thermodynamics gives
-\[T\nabla s =\nabla h-\frac{\nabla p}{\rho}\]\\
+{{NumEqn|<math>
+T\nabla s =\nabla h-\frac{\nabla p}{\rho}
+</math>}}
-\noindent Insert $\nabla p$ from the momentum equation\\
+Insert <math>\nabla p</math> from the momentum equation
-\[T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\]\\
+{{NumEqn|<math>
+T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}
+</math>}}
-\noindent Definition of total enthalpy ($h_o$)\\
+Definition of total enthalpy (<math>h_o</math>)
-\[h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)\]\\
+{{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>}}
-%\newpage
+The last term can be rewritten as
-\noindent 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
-\[\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}\]\\
+{{NumEqn|<math>
+\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}
+</math>}}
-\noindent which gives\\
+Insert <math>\nabla h</math> in the entropy equation gives
-\[\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}\]\\
+{{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>}}
-\noindent Insert $\nabla h$ in the entropy equation gives\\
+{{NumEqn|<math>
+T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}
-\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\
+</math>}}
-\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\

Crocco's equation: Difference between revisions

Latest revision as of 05:21, 1 April 2026

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