The compressible Euler equations are equations for perfect fluid. Perfect fluids have no heat conduction () and no viscosity (), so in the comoving frame the stress energy tensor is:

(we use ). Relativistic Euler equations are given by the conservation of the stress energy tensor and the particle number conservation:

By doing the nonrelativistic limit (see *Perfect Fluids* for a detailed
derivation), we get the following Euler equations:

where

is the total energy per unit volume, composed of the kinetic energy per unit volume () and the internal energy per unit volume (), where is the internal energy per unit mass (). The energy doesn’t contain the rest mass energy, but all other energies are hidden in the internal energy.

We use the ideal gas equations, so:

where is the number of moles of gas, is the molar mass of the gas (i.e. a mass of one mole of the gas, e.g. for dry air we get , as it is mainly composed of 20% of oxygen with atomic mass 16 and 78% of nitrogen with atomic mass , both form diatomic molecules, so the molecular mass is twice the atomic mass giving the total of , the rest is given by the other components and one also has to average over all isotopes), is the ideal gas constant ( is the Avogadro constant and is the Boltzmann constant), is the specific ideal gas constant (e.g. for dry air we get ), is the density of the gas (e.g. for dry air at the pressure and temperature we get ), is the specific heat capacity at constant volume (i.e. the amount of energy needed to heat one kg by one Kelvin at constant volume, e.g. for dry air the experimental value is about ), is the volume and is the temperature of the gas. Of those, , , , , are constants, , , and are functions of .

Here are the SI units of the various terms in the Euler equations:

In order to calculate the specific heat ratio , we use :

and the speed of sound is:

We choose 3 constants , and - characteristic length of the domain, velocity and density. Now we multiply the Euler equations with proper combinations of these constants as follows:

This is equal to:

where:

In particular, if , then

So the dimensionless Euler equations look exactly the same as the original ones, we just need to rescale all the quantities using the relations above.

We can write the Euler equations as:

where:

We solve for the unknowns , , , and as functions of , the rest (, , , , ) are either constants or depend on the unknowns. In order to convert from the physical quantities , , , , and to , ..., , we use:

the opposite conversion is:

Sometimes people also use , and instead of , and .

Note: , where is the fluid density current (it’s a 3-vector) and also (here is the same as , e.g. we are a bit sloppy about the notation), where is the density 4-current (e.g. the first 4 components of are exactly the components of the 4-current ):

where as usual is the relativistic index, is the speed of light, and in the nonrelativistic limit () we get and the remaining in will cancel with in , so it will not be present in the final equations (that involve terms like ). We can also just set as usual in relativistic physics.

The Euler equations:

are nonlinear. The time-derivative is approximated using the implicit Euler method

The vector-valued test functions for the above system of equations have the form:

After multiplying the equation system with the test functions and integrating
over the domain , we obtain (here the index is
numbering the 5 equations, so we are *not* summing over it):

Now we integrate by parts:

where is the outward surface normal to . Rearranging:

We can then linearize this for example by taking the flux jacobians on the previous time level .

The finite element formulation is obtained from here by replacing in the standard way the unknown solution by a piecewise-polynomial unknown function

where are the basis functions of the piecewise-polynomial finite element space. This turns the above weak formulation into a finite number of nonlinear algebraic equations of the form that will be solved using the Newton’s method.

We also derive the weak formulation for the explicit method. Euler equations:

The time-derivative is approximated using the explicit Euler method

The vector-valued test functions for the above system of equations have the form:

After multiplying the equation system with the test functions and integrating
over the domain , we obtain (here the index is
numbering the 5 equations, so we are *not* summing over it):

Now we integrate by parts:

where is the outward surface normal to . Rearranging:

Now we write the spatial derivatives using the so called flux Jacobians , and :

Similarly for and , so we get:

One nice thing about these particular , and functions is that they are homogeneous of degree 1:

so the Euler equation/formula for the homogeneous function is:

So both the and it’s derivative can be nicely factored out using the flux Jacobian:

by differentiating the first equation and substracting the second, we get:

similarly for and . To calculate the Jacobians, we’ll need:

then we can calculate the Jacobians (and we substitute for ):

where:

Discretizing the time derivative:

The vector-valued test functions for the above system of equations have the form:

After multiplying the equation system with the test functions and integrating over the domain , we obtain:

Now we integrate by parts:

where is the outward surface normal to . Rearranging:

The 2D flux Jacobians are:

In our 2D model we make the following assumptions:

and the boundary condition is as follows:

The weak formulation in 2D is (here ):

In order to specify the input forms for Hermes, we’ll write the weak formulation as:

where the forms are (we write instead of ):

In the last expression we do *not* sum over nor .
In particular:

We rewrite the boundary integral by rotating coordinates, so that the flow is only in the direction (thus we only have ):

Now we need to approximate somehow. We do that by solving the following 1D problem (Riemann problem):

or:

(1)

And we approximate . The initial condition is:

Now we write:

and substitute into (1):

so we get:

This is a nonlinear problem, that cannot be solved exactly. First, let doesn’t depend on . Then also are a constants:

and the solution is constant along the characteristic for and we get:

and

so:

In the nonlinear case we cannot solve it exactly, but we can approximate the solution by:

(2)

Let’s say the domain is for and we are applying the BC condition from . Then is taken from the solution and is prescribed, for example at the bottom it could be:

Now we need to calculate . First we write:

Explicit forms of the matrices:

For :

For :

In the boundary (line) integral we prescribe using a Dirichlet condition and calculate it at each iteration using:

where is a known function of time (it changes with the day and night) and also prescribe on the left and right end of the domain and at the top and bottom.

All the surface integrals turn out to be zero. On the top and bottom edges we have respectively and we prescribe , so we get (remember we do not sum over ):

where:

So all the components of the surface integral are zero, and for the test function is not there, because we prescribe the Dirichlet BC , so the surface integral vanishes for all .

Similarly on the left and right edges we have respectively and we prescribe , so we get (remember we do not sum over ):

where:

So all the components of the surface integral are zero, and for the test function is not there, because we prescribe the Dirichlet BC , so the surface integral vanishes for all .

The residual is:

where numbers the equations, numbers the finite element basis functions, , . The Jacobian is:

And the Newton method then is:

Author: Pavel Solin

(3)

where is the air density, is the velocity, , , is the temperature, , and is the gravitational acceleration constant. We use the perfect gas state equation for the pressure.

Boundary conditions are prescribed as follows:

- edge : , , ,
- edges : , , ,
- edge : , , ,

Initial conditions have the form

We will use the implicit Euler method in time, i.e.,

etc. Let’s discuss one equation of (3) at a time:

Continuity equation: The weak formulation of

reads

(4)

The global coefficient vector consists of four parts , , and corresponding to the fields , , and , respectively. The same holds for the vector function which consists of four parts , , and . Thus the global Jacobi matrix will have a four-by-four block structure. We denote

(5)

It follows from (4) and (5) that

First momentum equation: The second equation of (3) has the form

After applying the implicit Euler method, we obtain

Thus we obtain

Analogously,

Second momentum equation: The third equation of (3) reads

After applying the implicit Euler method, we obtain

Thus we obtain

Analogously,

Internal energy equation: The last equation of (3) has the form

where . This can be written equivalently as

Written in terms of single derivatives, this is

i.e.,

Weak formulation:

For the derivatives of the weak form we obtain: