4.1.11. Euler Momentum Equation¶
4.1.11.1. What is the derivation of the Euler Momentum Equation from the Cauchy Equation?¶
Cauchy Equation:
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) = \nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} + \rho \vec{g}\]
Inviscid flow \(\longrightarrow \overset{\underset{\mathrm{\rightrightarrows}}{}}{\tau}=0\):
\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} =
\begin{bmatrix}
\sigma_{11} & 0 & 0 \\
0 & \sigma_{22} & 0 \\
0 & 0 & \sigma_{33}
\end{bmatrix}\end{split}\]
Normal stresses = applied pressure:
\[\begin{split}\overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} = -p \delta_{ij}
= -p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I} =
\begin{bmatrix}
-p & 0 & 0 \\
0 & -p & 0 \\
0 & 0 & -p
\end{bmatrix}\end{split}\]
Expansion of divergence of shear stress:
\[\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{\sigma} =
\nabla \cdot (-p \overset{\underset{\mathrm{\rightrightarrows}}{}}{I}) =
-(\nabla \cdot \overset{\underset{\mathrm{\rightrightarrows}}{}}{I})p
-(\overset{\underset{\mathrm{\rightrightarrows}}{}}{I} \cdot \nabla)p = -\nabla p\]
Hence:
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) =
-\nabla p + \rho \vec{g}\]
4.1.11.2. What is the meaning of the terms in the Euler momentum equation?¶
\[\rho \left( {{\partial \vec{u}} \over {\partial t}}+ (\vec{u} \cdot \nabla) \vec{u} \right) =
-\nabla p + \rho \vec{g}\]
\(\rho {{\partial \vec{u}} \over {\partial t}}\) = Temporal change of momentum/unit volume at a fixed point
\(\rho(\vec{u} \cdot \nabla) \vec{u}\) = Spatial change of momentum/unit volume at a fixed point
\(-\nabla p\) = Pressure force/unit volume at a fixed point
\(\rho \vec{g}\) = Gravity force/unit volume at a fixed point
4.1.11.3. How are all the terms in the Euler Equations expanded in 3D?¶
x-direction:
\[\rho \left( {{\partial {u}} \over {\partial t}}+ {u} {{\partial u} \over {\partial x}} +
{v} {{\partial u} \over {\partial y}} + {w} {{\partial u} \over {\partial z}} \right)=
-{{\partial {p}} \over {\partial x}} + \rho g_x\]
y-direction:
\[\rho \left( {{\partial {v}} \over {\partial t}}+ {u} {{\partial v} \over {\partial x}} +
{v} {{\partial v} \over {\partial y}} + {w} {{\partial v} \over {\partial z}} \right)=
-{{\partial {p}} \over {\partial y}} + \rho g_y\]
z-direction:
\[\rho \left( {{\partial {w}} \over {\partial t}}+ {u} {{\partial w} \over {\partial x}} +
{v} {{\partial w} \over {\partial y}} + {w} {{\partial w} \over {\partial z}} \right)=
-{{\partial {p}} \over {\partial z}} + \rho g_z\]
Can also add source terms to the RHS \(S_x\), \(S_y\) and \(S_z\)