0
$\begingroup$

I'm looking at this page for the compressible Euler equations.

To me it seems, in the 1-dimensional case, there should be 3 unknowns: density, velocity, and pressure. This is because the energy $E$ is a function of density and velocity, as the internal energy $e$ seems like it should be constant, because the flow is adiabatic. Additionally, it is stated

The equations above thus represent conservation of mass, momentum, and energy. Mass density, Flow velocity and pressure are the so-called physical variables, while mass density, momentum density and total energy density are the so-called conserved variables.

So three unknowns, right?

However, later in the page, it is stated that

there are thus N+2 equations and N+3 unknowns

So there should be 3 equations and 4 unknowns. Thus an equation of state, relating pressure, energy, density, and velocity, is invoked, allowing us to solve for 4 unknowns with 4 equations.

However, before it appeared to me that there were only 3 unknowns. Is it that the internal energy $e$ is nonconstant, which gives us the fourth unknown, or is it something else?

$\endgroup$
1
$\begingroup$

The equations above thus represent conservation of mass, momentum, and energy. Mass density, Flow velocity and pressure are the so-called physical variables, while mass density, momentum density and total energy density are the so-called conserved variables.

So three unknowns, right?

Actually, there are four variables: density, velocity, pressure and total energy. Density, pressure and energy are scalars (the +3) and velocity is a vector of dimension $N$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.