Euler Equations (fluid Dynamics) - Conservation and Component Form

Conservation and Component Form

In differential form, the equations are:

$\begin{align} &{\partial\rho\over\partial t}+ \nabla\cdot(\rho\bold u)=0\\ &{\partial\rho{\bold u}\over\partial t}+ \nabla\cdot(\bold u\otimes(\rho \bold u))+\nabla p=\bold{0}\\ &{\partial E\over\partial t}+ \nabla\cdot(\bold u(E+p))=0, \end{align}$

where

ρ is the fluid mass density,
u is the fluid velocity vector, with components u, v, and w,
E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
p is the pressure,
denotes the tensor product, and
being the zero vector.

These equations may be expressed in subscript notation. The second equation includes the divergence of a dyadic product, and may be clearer in subscript notation:

${\partial\rho\over\partial t}+ \sum_{i=1}^3 {\partial(\rho u_i)\over\partial x_i} =0,$

${\partial(\rho u_j)\over\partial t}+ \sum_{i=1}^3 {\partial(\rho u_i u_j)\over\partial x_i}+ {\partial p\over\partial x_j} =0,$

${\partial E\over\partial t}+ \sum_{i=1}^3 {\partial((E+p) u_i)\over\partial x_i} =0,$

where the i and j subscripts label the three Cartesian components: ( x₁, x₂, x₃ ) = ( x, y, z ) and ( u₁, u₂, u₃ ) = ( u, v, w ). These equations may be more succinctly expressed using Einstein notation, in which matched indices imply a sum over those indices and and :

$\partial_t \rho+\partial_i(\rho u_i)=0\,$

$\partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p=0\,$

$\partial_t E+\partial_i((E+p)u_i)=0\,$

Note that the above equations are expressed in conservation form, as this format emphasizes their physical origins (and is often the most convenient form for computational fluid dynamics simulations). By subtracting the velocity times the mass conservation term, the second equation (momentum conservation), can also be expressed as:

$- u_j= \rho \partial_t u_j+\rho u_i \partial_i u_j+\partial_j p=0\,$

or, in vector notation:

$\rho\left( \frac{\partial}{\partial t}+{\bold u}\cdot\nabla \right){\bold u}+\nabla p=\bold{0}$

but this form for the momentum conservation equation obscures the direct connection between the Euler equations and Newton's second law of motion. Similarly, by subtracting the velocity times the above momentum conservation term, the third equation (energy conservation), can also be expressed as:

$\partial_t e+\partial_i(eu_i) + p\partial_i u_i=0\,$

$\frac{\partial e}{\partial t}+\nabla\cdot(e\bold u)+p\nabla\cdot \bold u=0$

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