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where the conservation quantity formula_36 in this case is a vector, and formula_37 is a flux matrix. This can be simply proved.
At last Euler equations can be recast into the particular equation:
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., formula_38 and formula_39) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics.
In convective form the incompressible Euler equations in case of density variable in space 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:
Here formula_36 has length formula_46 and formula_37 has size formula_48. In general (not only in the Froude limit) Euler equations are expressible as:
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:
The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form.
Mass density, flow velocity and pressure are the so-called "convective variables" (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called "conserved variables" (also called eulerian, or mathematical variables).
If one explicitates the material derivative the equations above are:
Coming back to the incompressible case, it now becomes apparent that the "incompressible constraint" typical of the former cases actually is a particular form valid for incompressible flows of the "energy equation", and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation:
Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.
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.
Since by definition the specific enthalpy is:
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.
In thermodynamics the independent variables are the specific volume, and the specific entropy, while the specific energy is a function of state of these two variables.
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:
In the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form:
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 "fundamental" equation of state contains all the thermodynamic information about the system (Callen, 1985), exactly like the couple of a "thermal" equation of state together with a "caloric" equation of state.
The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:
Here formula_36 has length N + 2 and formula_37 has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as:
We remark that also the Euler equation even when conservative (no external field, Froude limit) have no Riemann invariants in general. Some further assumptions are required
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:
Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form :
where formula_85 are called the flux Jacobians defined as the matrices:
Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians formula_85 are not functions of the state vector formula_36, the equations reveals "linear".
The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables.
In fact the tensor A is always diagonalizable. If the eigenvalues (the case of Euler equations) are all real the system is defined "hyperbolic", and physically eigenvalues represent the speeds of propagation of information. If they are all distinguished, the system is defined "strictly hyperbolic" (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.
If formula_89 is the right eigenvector of the matrix formula_90 corresponding to the eigenvalue formula_91, by building the projection matrix:
One can finally find the "characteristic variables" as:
Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields the characteristic equations:
The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables "w"i are called the "characteristic variables" and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it 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:
Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each "i"-th wave has shape "w""i""p""i" and speed of propagation "λ""i". In the following we show a very simple example of this solution procedure.
Waves in 1D inviscid, nonconductive thermodynamic fluid.
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) :
If one defines the vector of variables:
recalling that formula_69 is the specific volume, formula_101 the flow speed, formula_71 the specific entropy, the corresponding jacobian matrix is:
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.
This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific energy expressed as function of specific volume and specific entropy:
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:
The other two eigenvectors can be found with analogous procedure as:
Then the projection matrix can be built:
Finally it becomes apparent that the real parameter "a" previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the "wave speed". It remains to be shown that the sound speed corresponds to the particular case of an isentropic transformation:
Sound speed is defined as the wavespeed of an isentropic transformation:
by the definition of the isoentropic compressibility:
the soundspeed results always the square root of ratio between the isentropic compressibility and the density:
The sound speed in an ideal gas depends only on its temperature:
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:
Bernoulli's theorem is a direct consequence of the Euler equations.
The vector calculus identity of the cross product of a curl holds:
where the Feynman subscript notation formula_127 is used, which means the subscripted gradient operates only on the factor formula_37.
Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form:
the Euler momentum equation in Lamb's form becomes:
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:
And by projecting the momentum equation on the flow direction, i.e. along a "streamline", the cross product disappears because its result is always perpendicular to the velocity:
In the steady incompressible case the mass equation is simply:
that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes:
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:
so that the internal specific energy now features in the head.
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:
and the Bernoulli invariant for an inviscid gas flow is:
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.
In the usual case of small potential field, simply:
By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:
in the convective form of Euler momentum equation, one arrives to:
Friedmann deduced this equation for the particular case of a perfect gas and published it in 1922. However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.
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:
and by defining the specific total enthalpy: