Crocco's equation: Difference between revisions
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The momentum equation without body forces | The momentum equation without body forces | ||
<math | {{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 | {{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 | {{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 | {{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 | {{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 | {{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 | {{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 | {{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 | {{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>}} | ||
