Euler equations (fluid dynamics)

Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
Flow around a wing. This incompressible flow satisfies the Euler equations.

Although Euler first presented these equations in 1755, many fundamental questions about them remain unanswered.

Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy:

In the one dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers equation:

Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix):

Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.

The free Euler equations are conservative, in the sense they are equivalent to a conservation equation:

Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called conservation (or Eulerian) differential form, with vector notation:

Then incompressible Euler equations with uniform density have conservation variables:

Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3×3 and F has size 4×3, so the explicit forms are:

The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be:

but here the last term is identically zero for the incompressibility constraint.

The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:

The variables for the equations in conservation form are not yet optimised. In fact we could define:

In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation:

for an incompressible inviscid fluid the specific internal energy is constant along the flow lines

Basing on the mass conservation equation, one can put this equation in the conservation form:

meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.

The material derivative of the specific internal energy can be expressed as:

Then by substituting the momentum equation in this expression, one obtains:

And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation:

In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure.

Considering the first equation, variable must be changed from density to specific volume. By definition:

Then by substituting these expressions in the mass conservation equation:

This equation is the only belonging to general continuum equations, so only this equation have the same form for example also in Navier-Stokes equations.

On the other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:

since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited as:

It is convenient for brevity to switch the notation for the second order derivatives:

can be furtherly simplified in convective form by changing variable from specific energy to the specific entropy: in fact the first law of thermodynamics in local form can be written:

by substituting the material derivative of the internal energy, the energy equation becomes:

now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply:

For a thermodynamic fluid, the compressible Euler equations are consequently best written as:

for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines

meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.

On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy:

The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:

However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation:

Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as:

Another possible form for the energy equation, being particularly useful for isobarics, is:

Then the solution in terms of the original conservative variables is obtained by transforming back:

this computation can be explicited as the linear combination of the eigenvectors:

If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0) :

At first one must find the eigenvalues of this matrix by solving the characteristic equation:

This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements.

is defined positive. This statement corresponds to the two conditions:

The first condition is the one ensuring the parameter a is defined real.

Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system.

At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue λ1 one obtains:

Basing on the third equation that simply has solution s1=0, the system reduces to:

The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:

Sound speed is defined as the wavespeed of an isentropic transformation:

the soundspeed results always the square root of ratio between the isentropic compressibility and the density:

In an ideal gas the isoentropic transformation is described by the Poisson's law:

Then by substitution in the general definitions for an ideal gas the isentropic compressibility is simply proportional to the pressure:

Notably, for an ideal gas the ideal gas law holds, that in mathematical form is simply:

By substituting this ratio in the Newton–Laplace law, the expression of the sound speed into an ideal gas as function of temperature is finally achieved.

Since the specific enthalpy in an ideal gas is proportional to its temperature:

the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:

the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows:

In fact, in case of an external conservative field, by defining its potential φ:

In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:

that is . Then the Euler momentum equation in the steady incompressible case becomes:

the mass conservation for a steady incompressible flow states that the density along a streamline is constant

The convenience of defining the total head for an inviscid liquid flow is now apparent:

That is, .

the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant

In the most general steady (compressibile) case the mass equation in conservation form is:

The right-hand side appears on the energy equation in convective form, which on the steady state reads:

Since the external field potential is usually small compared to the other terms, it is convenient to group the latter ones in the total enthalpy:

That is, .

the energy balance for a steady inviscid flow in an external conservative field states that the sum of the total enthalpy and the external potential is constant along a streamline

By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:

On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:

In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form:

From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.

Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:

Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux:

For one-dimensional Euler equations the conservation variables and the flux are the vectors:

Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:

The energy equation is an integral form of the Bernoulli equation in the compressible case. The former mass and momentum equations by substitution lead to the Rayleigh equation:

One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.

The Hugoniot equation, coupled with the fundamental equation of state of the material:

allowing to quantify deviations from the Hugoniot equation, similarly to the previous definition of the hydraulic head, useful for the deviations from the Bernoulli equation.

In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:

Now inverting the equation for temperature T(e) we deduce that for an ideal polytropic gas the isochoric heat capacity is a constant:

and similarly for an ideal polytropic gas the isobaric heat capacity results constant:

The specific enthalpy results by substitution of the latter and of the ideal gas law:

From this equation one can derive the equation for pressure by its thermodynamic definition:

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:

The third equation expresses that pressure is constant along the binormal axis.

The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.