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 $$S_z$$