Governing equations on differential form: Difference between revisions
From Flowpedia
No edit summary |
No edit summary |
||
| Line 48: | Line 48: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | ||
</math>}} | </math>|label=eq-cont-pde}} | ||
which is the continuity equation on partial differential form. | which is the continuity equation on partial differential form. | ||
| Line 86: | Line 86: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f} | \frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f} | ||
</math>}} | </math>|label=eq-mom-pde}} | ||
which is the momentum equation on partial differential form | which is the momentum equation on partial differential form | ||
| Line 120: | Line 120: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>|label=eq-energy-pde}} | ||
which is the energy equation on partial differential form | which is the energy equation on partial differential form | ||
| Line 148: | Line 148: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla | \frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla | ||
</math>}} | </math>|label=eq-cont-pde-non-cons}} | ||
where the first term of the right hand side is the local derivative and the second term is the convective derivative. | where the first term of the right hand side is the local derivative and the second term is the convective derivative. | ||
| Line 204: | Line 204: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | \frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | ||
</math>}} | </math>|label=eq-mom-pde-non-cons}} | ||
==== Conservation of Energy ==== | ==== Conservation of Energy ==== | ||
The last equation on non-conservation differential form is the energy equation. We start by rewriting the energy equation on differential form | The last equation on non-conservation differential form is the energy equation. We start by rewriting the energy equation on differential form {{EquationNote|label=eq-energy-pde}}, repeated here for convenience | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>|nonumber=1}} | ||
Total enthalpy, <math>h_o</math>, is replaced with total energy, <math>e_o</math> | Total enthalpy, <math>h_o</math>, is replaced with total energy, <math>e_o</math> | ||
| Line 229: | Line 229: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\rho\frac{\partial e_o}{\partial t} + e_o\frac{\partial \rho}{\partial t} + \rho\mathbf{v}\cdot\nabla e_o + e_o\nabla\cdot(\rho \mathbf{v}) + \nabla\cdot(p\mathbf{v})= \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \rho\frac{\partial e_o}{\partial t} + e_o\frac{\partial \rho}{\partial t} + \rho\mathbf{v}\cdot\nabla e_o + e_o\nabla\cdot(\rho \mathbf{v}) + \nabla\cdot(p\mathbf{v})=</math><br><br><math>= \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>}} | ||
| Line 242: | Line 242: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>|label=eq-energy-pde-non-cons}} | ||
==== Summary ==== | ==== Summary ==== | ||
| Line 274: | Line 274: | ||
</math>}} | </math>}} | ||
Inserted in | Inserted in {{EquationNote|label=eq-energy-pde-non-cons|nopar=1}}, this gives | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 280: | Line 280: | ||
</math>}} | </math>}} | ||
Now, let's replace the substantial derivative <math>D\mathbf{v}/Dt</math> using the momentum equation on non-conservation form | Now, let's replace the substantial derivative <math>D\mathbf{v}/Dt</math> using the momentum equation on non-conservation form {{EquationNote|label=eq-mom-pde-non-cons}}. | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 289: | Line 289: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\rho\frac{De}{Dt} \cancel{-\mathbf{v}\cdot\nabla p} + \cancel{\mathbf{v}\cdot\nabla p} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho\Rightarrow \rho\frac{De}{Dt} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho | \rho\frac{De}{Dt} \cancel{-\mathbf{v}\cdot\nabla p} + \cancel{\mathbf{v}\cdot\nabla p} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho\Rightarrow</math><br><br><math>\Rightarrow\rho\frac{De}{Dt} + p(\nabla\cdot\mathbf{v}) = \dot{q}\rho | ||
</math>}} | </math>}} | ||
| Line 296: | Line 296: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{De}{Dt} + \frac{p}{\rho}(\nabla\cdot\mathbf{v}) = \dot{q} | \frac{De}{Dt} + \frac{p}{\rho}(\nabla\cdot\mathbf{v}) = \dot{q} | ||
</math>}} | </math>|label=eq-energy-pde-non-cons-b}} | ||
Conservation of mass gives | Conservation of mass gives | ||
| Line 304: | Line 304: | ||
</math>}} | </math>}} | ||
Insert in | Insert in {{EquationNote|label=eq-energy-pde-non-cons-b|nopar=1}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 322: | Line 322: | ||
</math>}} | </math>}} | ||
with <math>De/Dt</math> from | with <math>De/Dt</math> from {{EquationNote|label=eq-energy-pde-non-cons-b|nopar=1}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 330: | Line 330: | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{Dh}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt} | \frac{Dh}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt} | ||
</math>}} | </math>|label=eq-energy-pde-non-cons-c}} | ||
==== Total Enthalpy Formulation ==== | ==== Total Enthalpy Formulation ==== | ||
| Line 338: | Line 338: | ||
</math>}} | </math>}} | ||
From the momentum equation | From the momentum equation {{EquationNote|label=eq-mom-pde-non-cons}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
| Line 350: | Line 350: | ||
</math>}} | </math>}} | ||
Inserting <math>Dh/Dt</math> from | Inserting <math>Dh/Dt</math> from {{EquationNote|label=eq-energy-pde-non-cons-c|nopar=1}} gives | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{Dh_o}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}+\mathbf{v}\cdot\mathbf{f} -\frac{1}{\rho}\mathbf{v}\cdot\nabla p = \frac{1}{\rho}\left[\frac{Dp}{Dt}-\mathbf{v}\cdot\nabla p\right] + \dot{q} + \mathbf{v}\cdot\mathbf{f} | \frac{Dh_o}{Dt}=\dot{q} + \frac{1}{\rho}\frac{Dp}{Dt}+\mathbf{v}\cdot\mathbf{f} -\frac{1}{\rho}\mathbf{v}\cdot\nabla p =</math><br><br><math>=\frac{1}{\rho}\left[\frac{Dp}{Dt}-\mathbf{v}\cdot\nabla p\right] + \dot{q} + \mathbf{v}\cdot\mathbf{f} | ||
</math>}} | </math>}} | ||
