In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively (the simplest form for Euler equations being the conservation of the specific entropy). Historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – of the more general compressible equations together as "the Euler equations".
From the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field (i.e., in the limit of high Froude number). In fact, like any Cauchy equation, the Euler equations originally formulated in convective form (also called "Lagrangian form") can also be put in the "conservation form" (also called "Eulerian form"). The conservation form emphasizes the mathematical interpretation of the equations as conservation equations through a control volume fixed in space, and is the most important for these equations also from a numerical point of view. The convective form emphasizes changes to the state in a frame of reference moving with the fluid.

History

The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in ''Mémoires de l'Académie des Sciences de Berlin'' in 1757 (in this article Euler actually published only the ''general'' form of the continuity equation and the momentum equation; the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816. During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector

Incompressible Euler equations with constant and uniform density

In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are: where: *$\backslash mathbf\; u$ is the flow velocity vector, with components in an ''N''-dimensional space $u\_1,\; u\_2,\; \backslash dots,\; u\_N$, *$=+\backslash mathbf\; u\backslash cdot\backslash nabla\backslash mathbf\; v$, for a generic function (or field) $\backslash mathbf\; v$ denotes its material derivative in time with respect to the advective field $\backslash mathbf\; u$ and *$\backslash nabla$ denotes the gradient with respect to space, *$\backslash cdot$ denotes the scalar product, *$\backslash nabla$ is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and *$\backslash nabla\; \backslash cdot\; \backslash mathbf\; u$ is the flow velocity divergence (second equation), *$w$ is the specific (with the sense of ''per unit mass'') thermodynamic work, the internal source term. * $\backslash mathbf$ represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on. The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become: :$\backslash left\backslash \; Euler\; equations\; in\; the\; Froude\; limit\; (no\; external\; field)\; are\; named\; free\; equations\; and\; are\; conservative.\; The\; limit\; of\; high\; Froude\; numbers\; (low\; external\; field)\; is\; thus\; notable\; and\; can\; be\; studied\; withperturbation\; theory.$

Conservation form

The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods. The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: :$\backslash frac+\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =,$ or simply in Einstein notation: :$\backslash frac+\; \backslash frac=\; 0\_i,$ where the conservation quantity $\backslash mathbf\; y$ in this case is a vector, and $\backslash mathbf\; F$ is a flux matrix. This can be simply proved. At last Euler equations can be recast into the particular equation:

Spatial dimensions

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., $x$ and $t$) 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.

Incompressible Euler equations

In convective form the incompressible Euler equations in case of density variable in space are: where the additional variables are: *$\backslash rho$ is the fluid mass density, *$p$ is the pressure, $p\; =\; \backslash rho\; w$. The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be: :$+\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; +\; \backslash rho\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u\; =\; 0$ but here the last term is identically zero for the incompressibility constraint.

Conservation form

The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: :$=\backslash begin\backslash rho\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; u\; \backslash \backslash 0\backslash end;\; \backslash qquad\; =\backslash begin\backslash rho\; \backslash mathbf\; u\backslash \backslash \backslash rho\; \backslash mathbf\; u\; \backslash otimes\; \backslash mathbf\; u\; +\; p\; \backslash mathbf\; I\backslash \backslash \backslash mathbf\; u\backslash end.$ Here $\backslash mathbf\; y$ has length $N+2$ and $\backslash mathbf\; F$ has size $(N+2)N$. In general (not only in the Froude limit) Euler equations are expressible as: :$\backslash frac\; \backslash begin\backslash rho\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; u\; \backslash \backslash 0\backslash end+\; \backslash nabla\; \backslash cdot\; \backslash begin\backslash rho\; \backslash mathbf\; u\backslash \backslash \backslash rho\; \backslash mathbf\; u\; \backslash otimes\; \backslash mathbf\; u\; +\; p\; \backslash mathbf\; I\backslash \backslash \; \backslash mathbf\; u\backslash end\; =\; \backslash begin0\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; g\; \backslash \backslash \; 0\; \backslash end$

Conservation variables

The variables for the equations in conservation form are not yet optimised. In fact we could define: :$=\backslash begin\backslash rho\; \backslash \backslash \; \backslash mathbf\; j\; \backslash \backslash 0\backslash end;\; \backslash qquad\; =\backslash begin\; \backslash mathbf\; j\; \backslash \backslash \; \backslash frac\; \backslash rho\; \backslash ,\; \backslash mathbf\; j\; \backslash otimes\; \backslash mathbf\; j+\; p\; \backslash mathbf\; I\backslash \backslash \; \backslash frac\; \backslash mathbf\; j\; \backslash rho\; \backslash end.$ where: * $\backslash mathbf\; j\; =\; \backslash rho\; \backslash mathbf\; u$ is the momentum density, a conservation variable. where: * $\backslash mathbf\; f\; =\; \backslash rho\; \backslash mathbf\; g$ is the force density, a conservation variable.

Euler equations

In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation: &= -\frac + \mathbf \\.2ex &= -\frac\nabla \cdot \mathbf \end\right. |cellpadding |border |border colour = #FF0000 |background colour = #ECFCF4 where the additional variables here is: *$e$ is the specific internal energy (internal energy per unit mass). 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: :$\backslash left\backslash \}\; For\; a\; thermodynamic\; fluid,\; the\; compressible\; Euler\; equations\; are\; consequently\; best\; written\; as:\; \&=\; ve\_\backslash nabla\; v\; +\; ve\_\backslash nabla\; s\; +\; \backslash mathbf\; \backslash \backslash .2ex\&=\; 0\; \backslash end\backslash right.$ |cellpadding |border |border colour = #FFFF00 |background colour = #ECFCF4 where: * $v$ is the specific volume * $\backslash mathbf\; u$ is the flow velocity vector * $s$ is the specific entropy 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: :$+\; \backslash nabla\; \backslash cdot\; (\backslash rho\; s\; \backslash mathbf\; u)\; =\; 0$ 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: :$e\; =\; e(v,\; s)$ 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.

Conservation form

The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: :$=\; \backslash begin\; \backslash rho\; \backslash \backslash \; \backslash mathbf\; j\; \backslash \backslash \; E^t\; \backslash end;\; \backslash qquad\; =\; \backslash begin\; \backslash mathbf\; j\; \backslash \backslash \; \backslash frac\; 1\; \backslash rho\; \backslash mathbf\; j\; \backslash otimes\; \backslash mathbf\; j\; +\; p\; \backslash mathbf\; I\; \backslash \backslash \; \backslash left(E^t\; +\; p\backslash right)\; \backslash frac\backslash mathbf\; \backslash end.$ where: * $\backslash mathbf\; j\; =\; \backslash rho\; \backslash mathbf\; u$ is the momentum density, a conservation variable. * $E^t\; =\; \backslash rho\; e\; +\; \backslash frac\; \backslash rho\; u^2$ is the total energy density (total energy per unit volume). Here $\backslash mathbf\; y$ has length N + 2 and $\backslash mathbf\; F$ has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as: where: * $\backslash mathbf\; f\; =\; \backslash rho\; \backslash mathbf\; g$ is the force density, a conservation variable. 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: :$(\backslash rho\; s)\; +\; \backslash nabla\; \backslash cdot\; (\backslash rho\; s\; \backslash mathbf\; u)\; =\; 0$ Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as: \end = \begin0 \\ \mathbf \\ 0 \end |cellpadding |border |border colour = #FFFF00 |background colour = #ECFCF4 where: * $S\; =\; \backslash rho\; s$ is the entropy density, a thermodynamic conservation variable. Another possible form for the energy equation, being particularly useful for isobarics, is: :$\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash left(H^t\; \backslash mathbf\; u\backslash right)\; =\; \backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; f\; -\; \backslash frac$ where: *$H^t\; =\; E^t\; +\; p\; =\; \backslash rho\; e\; +\; p\; +\; \backslash frac\; \backslash rho\; u^2$ is the total enthalpy density.

Quasilinear form and characteristic equations

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 : :$\backslash frac\; +\; \backslash mathbf\; A\_i\; \backslash frac\; =\; .$ where $\backslash mathbf\; A\_i$ are called the flux Jacobians defined as the matrices: :$\backslash mathbf\; A\_i\; (\backslash mathbf\; y)=\backslash frac.$ Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians $\backslash mathbf\; A\_i$ are not functions of the state vector $\backslash mathbf\; y$, the equations reveals ''linear''.

Characteristic equations

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 $\backslash mathbf\; p\_i$ is the right eigenvector of the matrix $\backslash mathbf\; A$ corresponding to the eigenvalue $\backslash lambda\_i$, by building the projection matrix: :$\backslash mathbf\; =\; \backslash leftmathbf\_1,\; \backslash mathbf\_2,\; ...,\; \backslash mathbf\_n\backslash right/math>\; One\; can\; finally\; find\; the\; \text{'}\text{'}characteristic\; variables\text{'}\text{'}\; as:\; :$ \backslash mathbf=\; \backslash mathbf^\backslash mathbf,$Since\; A\; is\; constant,\; multiplying\; the\; original\; 1-D\; equation\; in\; flux-Jacobian\; form\; with\; P-1yields\; the\; characteristic\; equations:\; :$ \backslash frac\; +\; \backslash lambda\_j\; \backslash frac\; =\; 0\_i$The\; original\; equations\; have\; beendecoupledinto\; N+2\; characteristic\; equations\; each\; describing\; a\; simple\; wave,\; with\; the\; eigenvalues\; being\; the\; wave\; speeds.\; The\; variables\; \text{'}\text{'}w\text{'}\text{'}$_{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:
:$w\_i(x,\; t)\; =\; w\_i\backslash left(x\; -\; \backslash lambda\_i\; t,\; 0\backslash right)$
Then the solution in terms of the original conservative variables is obtained by transforming back:
:$\backslash mathbf\; =\; \backslash mathbf\; \backslash mathbf,$
this computation can be explicited as the linear combination of the eigenvectors:
:$\backslash mathbf(x,\; t)\; =\; \backslash sum\_^m\; w\_i\backslash left(x\; -\; \backslash lambda\_i\; t,\; 0\backslash right)\; \backslash mathbf\; p\_i,$
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) : :$\backslash left\backslash \{\backslash begin\{align\}\; \{\backslash partial\; v\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; v\; \backslash over\; \backslash partial\; x\}\; -\; v\; \{\backslash partial\; u\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash \backslash .2ex\{\backslash partial\; u\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; u\; \backslash over\; \backslash partial\; x\}\; -\; e\_\{vv\}\; v\; \{\backslash partial\; v\; \backslash over\; \backslash partial\; x\}\; -\; e\_\{vs\}v\; \{\backslash partial\; s\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash \backslash .2ex\{\backslash partial\; s\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; s\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash end\{align\}\backslash right.$ If one defines the vector of variables: :$\backslash mathbf\{y\}\; =\; \backslash begin\{pmatrix\}v\; \backslash \backslash \; u\; \backslash \backslash \; s\backslash end\{pmatrix\}$ recalling that $v$ is the specific volume, $u$ the flow speed, $s$ the specific entropy, the corresponding jacobian matrix is: : $\{\backslash mathbf\; A\}=\backslash begin\{pmatrix\}u\; \&\; -v\; \&\; 0\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u\; \&\; -\; e\_\{vs\}\; v\; \backslash \backslash \; 0\; \&\; 0\; \&\; u\; \backslash end\{pmatrix\}.$ At first one must find the eigenvalues of this matrix by solving the characteristic equation: : $\backslash det(\backslash mathbf\; A(\backslash mathbf\; y)\; -\; \backslash lambda(\backslash mathbf\; y)\; \backslash mathbf\; I)\; =\; 0$ that is explicitly: : $\backslash det\backslash begin\{bmatrix\}u-\backslash lambda\; \&\; -v\; \&\; 0\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u-\backslash lambda\; \&\; -\; e\_\{vs\}\; v\; \backslash \backslash \; 0\; \&\; 0\; \&\; u-\backslash lambda\; \backslash end\{bmatrix\}=0$ This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements. : $(u-\backslash lambda)\; \backslash det\; \backslash begin\{bmatrix\}u-\backslash lambda\; \&\; -v\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u\; -\backslash lambda\; \backslash end\{bmatrix\}=0$ Now by computing the determinant 2×2: : $(u\; -\; \backslash lambda)\backslash left((u\; -\; \backslash lambda)^2\; -\; e\_\{vv\}\; v^2\backslash right)\; =\; 0$ by defining the parameter: : $a(v,s)\; \backslash equiv\; v\; \backslash sqrt\; \{e\_\{vv$ or equivalently in mechanical variables, as: : $a(\backslash rho,p)\; \backslash equiv\; \backslash sqrt\; \{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}$ 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: : $\backslash begin\{pmatrix\}e\_\{vv\}\; \&\; e\_\{vs\}\; \backslash \backslash \; e\_\{vs\}\; \&\; e\_\{ss\}\; \backslash end\{pmatrix\}$ is defined positive. This statement corresponds to the two conditions: :$\backslash left\backslash \{\backslash begin\{align\}\; e\_\{vv\}\; \&>\; 0\; \backslash \backslash .2exe\_\{vv\}e\_\{ss\}\; -\; e\_\{vs\}^2\; \&>\; 0\; \backslash end\{align\}\backslash right.$ The first condition is the one ensuring the parameter ''a'' is defined real. The characteristic equation finally results: : $(u\; -\; \backslash lambda)\backslash left((u\; -\; \backslash lambda)^2\; -\; a^2\backslash right)\; =\; 0$ That has three real solutions: : $\backslash lambda\_1(v,u,s)\; =\; u-a(v,s)\; \backslash quad\; \backslash lambda\_2(u)=\; u,\; \backslash quad\; \backslash lambda\_3(v,u,s)\; =\; u+a(v,s)$ 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 λ_{1} one obtains:
: $\backslash begin\{pmatrix\}a\; \&\; -v\; \&\; 0\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; a\; \&\; -\; e\_\{vs\}\; v\; \backslash \backslash \; 0\; \&\; 0\; \&\; a\; \backslash end\{pmatrix\}\; \backslash begin\{pmatrix\}v\_1\backslash \backslash \; u\_1\; \backslash \backslash s\_1\; \backslash end\{pmatrix\}=0$
Basing on the third equation that simply has solution s_{1}=0, the system reduces to:
: $\backslash begin\{pmatrix\}a\; \&\; -v\; \backslash \backslash -a^2\; /v\&\; a\; \backslash end\{pmatrix\}\; \backslash begin\{pmatrix\}v\_1\backslash \backslash \; u\_1\; \backslash end\{pmatrix\}=0$
The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:
: $\backslash mathbf\; p\_1=\backslash begin\{pmatrix\}v\backslash \backslash \; a\; \backslash \backslash 0\backslash end\{pmatrix\}$
The other two eigenvectors can be found with analogous procedure as:
: $\backslash mathbf\; p\_2=\backslash begin\{pmatrix\}\; e\_\{vs\}\; \backslash \backslash \; 0\backslash \backslash \; -\; \backslash left(\backslash frac\; a\; v\; \backslash right)^2\; \backslash end\{pmatrix\},\; \backslash qquad\; \backslash mathbf\; p\_3=\backslash begin\{pmatrix\}v\backslash \backslash \; -a\; \backslash \backslash 0\backslash end\{pmatrix\}$
Then the projection matrix can be built:
: $\backslash mathbf\; P\; (v,u,s)=(\; \backslash mathbf\{p\}\_1,\; \backslash mathbf\{p\}\_2,\; \backslash mathbf\{p\}\_3)\; =\backslash begin\{pmatrix\}\; v\; \&\; e\_\{vs\}\; \&\; v\backslash \backslash \; a\; \&\; 0\; \&\; -a\; \backslash \backslash \; 0\; \&\; -\; \backslash left(\backslash frac\; a\; v\; \backslash right)^2\; \&\; 0\; \backslash end\{pmatrix\}$
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:
:$a\_s\; \backslash equiv\; \backslash sqrt\; \{\backslash left(\{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}\; \backslash right)\_s\}$

Compressibility and sound speed

Sound speed is defined as the wavespeed of an isentropic transformation: :$a\_s(\backslash rho,p)\; \backslash equiv\; \backslash sqrt\; \{\backslash left(\{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}\; \backslash right)\_s\}$ by the definition of the isoentropic compressibility: :$K\_s\; (\backslash rho,p)\; \backslash equiv\; \backslash frac\; 1\; \backslash rho\; \backslash left(\{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}\; \backslash right)\_s$ the soundspeed results always the square root of ratio between the isentropic compressibility and the density: :$a\_s\; \backslash equiv\; \backslash sqrt\; \{\backslash frac\; \{K\_s\}\; \backslash rho\}$

Ideal gas

The sound speed in an ideal gas depends only on its temperature: :$a\_s\; (T)\; =\; \backslash sqrt\; \{\backslash gamma\; \backslash frac\; T\; m\}$ Since the specific enthalpy in an ideal gas is proportional to its temperature: :$h\; =\; c\_p\; T\; =\; \backslash frac\; \{\backslash gamma\}\{\backslash gamma-1\}\; \backslash frac\; T\; m$ the sound speed in an ideal gas can also be made dependent only on its specific enthalpy: :$a\_s\; (h)\; =\; \backslash sqrt\; \{(\backslash gamma\; -1)\; h\}$

Bernoulli's theorem for steady inviscid flow

Bernoulli's theorem is a direct consequence of the Euler equations.

Incompressible case and Lamb's form

The vector calculus identity of the cross product of a curl holds: :$\backslash mathbf\{v\; \backslash \; \backslash times\; \}\; \backslash left(\; \backslash mathbf\{\; \backslash nabla\; \backslash times\; F\}\; \backslash right)\; =\; \backslash nabla\_F\; \backslash left(\; \backslash mathbf\{v\; \backslash cdot\; F\; \}\; \backslash right)\; -\; \backslash mathbf\{v\; \backslash cdot\; \backslash nabla\; \}\; \backslash mathbf\{\; F\}\; \backslash \; ,$ where the Feynman subscript notation $\backslash nabla\_F$ is used, which means the subscripted gradient operates only on the factor $\backslash mathbf\; F$. 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash mathbf\; u\; =\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; \backslash times\; \backslash mathbf\; u$ the Euler momentum equation in Lamb's form becomes: :$\backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; \backslash times\; \backslash mathbf\{u\}\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}\; =\; \backslash mathbf\{g\}\; =\; \backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; -\; \backslash mathbf\{u\}\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}$ Now, basing on the other identity: :$\backslash nabla\; \backslash left(\; \backslash frac\; \{p\}\{\backslash rho\}\; \backslash right)\; =\; \backslash frac\; \{\backslash nabla\; p\}\{\backslash rho\}\; -\; \backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho$ the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows: :$\backslash nabla\; \backslash left(\backslash frac\{1\}\{2\}u^2\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; -\; \backslash mathbf\; g\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; -\; \backslash frac\{\backslash partial\; \backslash mathbf\; u\}\{\backslash partial\; t\}$ In fact, in case of an external conservative field, by defining its potential φ: :$\backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; -\; \backslash frac\{\backslash partial\; \backslash mathbf\; u\}\{\backslash partial\; t\}$ In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes: :$\backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)$ 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\backslash frac\{1\}\{2\}u^2\; +\; \backslash phi\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\backslash rho$ In the steady incompressible case the mass equation is simply: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; =\; 0$, 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; 0$ The convenience of defining the total head for an inviscid liquid flow is now apparent: :$b\_l\; \backslash equiv\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; ,$ which may be simply written as: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; b\_l\; =\; 0$ 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.

Compressible case

In the most general steady (compressibile) case the mass equation in conservation form is: :$\backslash nabla\; \backslash cdot\; \backslash mathbf\; j\; =\; \backslash rho\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u\; +\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; =\; 0$ . Therefore, the previous expression is rather :$\backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; \backslash left(\{\backslash frac\{1\}\{2u^2\; +\; \backslash phi\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; =\; \backslash frac\{p\}\{\backslash rho\}\backslash nabla\; \backslash cdot\; \backslash mathbf\{u\}$ The right-hand side appears on the energy equation in convective form, which on the steady state reads: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; e\; =\; -\; \backslash frac\{p\}\{\backslash rho\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u$ The energy equation therefore becomes: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\; e\; +\; \backslash frac\; p\; \backslash rho\; +\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; \backslash right)\; =\; 0,$ 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: :$h^t\; \backslash equiv\; e\; +\; \backslash frac\; p\; \backslash rho\; +\; \backslash frac\; 1\; 2\; u^2$ and the Bernoulli invariant for an inviscid gas flow is: :$b\_g\; \backslash equiv\; h^t\; +\; \backslash phi\; =\; b\_l\; +\; e\; ,$ which can be written as: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; b\_g\; =\; 0$ 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; h^t\; \backslash sim\; 0$

Friedmann form and Crocco form

By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form: :$v\; \backslash nabla\; p\; =\; -T\; \backslash nabla\; s\; +\; \backslash nabla\; h$ in the convective form of Euler momentum equation, one arrives to: :$\backslash frac\{D\backslash mathbf\; u\}\{Dt\}=T\; \backslash nabla\backslash ,s-\backslash nabla\; \backslash ,h$ 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: :$\backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\; \backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; \backslash times\; \backslash mathbf\{u\}\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}\; =\; \backslash mathbf\{g\}$ and by defining the specific total enthalpy: :$h^t\; =\; h\; +\; \backslash frac\{1\}\{2\}u^2$ one arrives to the Crocco–Vazsonyi form (Crocco, 1937) of the Euler momentum equation: :$\backslash frac\{\backslash partial\; \backslash mathbf\{\; u\{\backslash partial\; t\}\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; \backslash times\; \backslash mathbf\; u\; -\; T\; \backslash nabla\; s\; +\; \backslash nabla\; h^t\; =\; \backslash mathbf\{g\}$ 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: :$\backslash left\backslash \{\backslash begin\{align\}\; \backslash mathbf\{u\}\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{u\}\; +\; T\backslash nabla\; s\; -\; \backslash nabla\; h^t\; \&=\; \backslash mathbf\{g\}\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; s\; \&=\; 0\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; h^t\; \&=\; 0\; \backslash end\{align\}\backslash right.$ Finally if the flow is also isothermal: :$T\; \backslash nabla\; s\; =\; \backslash nabla\; (T\; s)$ by defining the specific total Gibbs free energy: :$g^t\; \backslash equiv\; h^t\; +\; Ts$ the Crocco's form can be reduced to: :$\backslash left\backslash \{\backslash begin\{align\}\; \backslash mathbf\{u\}\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{u\}\; -\; \backslash nabla\; g^t\; \&=\; \backslash mathbf\{g\}\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; g^t\; \&=\; 0\; \backslash end\{align\}\backslash right.$ From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.

Discontinuities

The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot equations. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See Navier–Stokes equations) Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur. To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the ''local'' forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are ''local variables'') of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.

Rankine–Hugoniot equations

Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain: :$\backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =\; \backslash mathbf\; 0$ where in general F is the flux matrix. By integrating this local equation over a fixed volume V_{m}, it becomes:
:$\backslash int\_\{V\_m\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; dV\; =\; \backslash mathbf\; 0.$
Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux:
:$\backslash oint\_\{\backslash partial\; V\_m\}\; \backslash mathbf\; F\; ds\; =\; \backslash mathbf\; 0.$
This ''global form'' simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus:
:$\backslash int\_\{x\_m\}^\{x\_\{m+1\; \backslash mathbf\; F\backslash left(x\text{'}\backslash right)\; dx\text{'}\; =\; \backslash mathbf\; 0,$
that is the simple finite difference equation, known as the ''jump relation'':
:$\backslash Delta\; \backslash mathbf\; F\; =\; \backslash mathbf\; 0.$
That can be made explicit as:
:$\backslash mathbf\; F\_\{m+1\}\; -\; \backslash mathbf\; F\_m\; =\; \backslash mathbf\; 0$
where the notation employed is:
:$\backslash mathbf\; F\_\{m\}\; =\; \backslash mathbf\; F(x\_m).$
Or, if one performs an indefinite integral:
:$\backslash mathbf\; F\; -\; \backslash mathbf\; F\_0\; =\; \backslash mathbf\; 0.$
On the other hand, a transient conservation equation:
:$\{\backslash partial\; y\; \backslash over\; \backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =\; \backslash mathbf\; 0$
brings to a jump relation:
:$\backslash frac\{dx\}\{dt\}\; \backslash ,\; \backslash Delta\; u\; =\; \backslash Delta\; \backslash mathbf\; F.$
For one-dimensional Euler equations the conservation variables and the flux are the vectors:
:$\backslash mathbf\; y\; =\; \backslash begin\{pmatrix\}\; \backslash frac\{1\}\{v\}\; \backslash \backslash \; j\; \backslash \backslash \; E^t\; \backslash end\{pmatrix\},$
:$\backslash mathbf\; F\; =\; \backslash begin\{pmatrix\}\; j\; \backslash \backslash \; v\; j^2\; +\; p\; \backslash \backslash \; v\; j\; \backslash left(E^t\; +\; p\backslash right)\; \backslash end\{pmatrix\},$
where:
* $v$ is the specific volume,
* $j$ is the mass flux.
In the one dimensional case the correspondent jump relations, called the Rankine–Hugoniot equations, are:<
: $\backslash left\backslash \{\backslash begin\{align\}\; \backslash frac\{dx\}\{dt\}\backslash Delta\; \backslash left(\backslash frac\{1\}\{v\}\backslash right)\; \&=\; \backslash Delta\; j\backslash \backslash .2ex\backslash frac\{dx\}\{dt\}\backslash Delta\; j\; \&=\; \backslash Delta\backslash left(vj^2\; +\; p\backslash right)\backslash \backslash .2ex\backslash frac\{dx\}\{dt\}\backslash Delta\; E^t\; \&=\; \backslash Delta\; \backslash left(jv\backslash left(E^t\; +\; p\backslash right)\backslash right)\; \backslash end\{align\}\backslash right.\; .$
In the steady one dimensional case the become simply:
: $\backslash left\backslash \{\backslash begin\{align\}\; \backslash Delta\; j\; \&=\; 0\backslash \backslash .2ex\backslash Delta\backslash left(v\; j^2\; +\; p\backslash right)\; \&=\; 0\; \backslash \backslash .2ex\backslash Delta\backslash left(j\backslash left(\backslash frac\{E^t\}\{\backslash rho\}\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\backslash right)\; \&=\; 0\; \backslash end\{align\}\backslash right.\; .$
Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:
:$\backslash left\backslash \{\backslash begin\{align\}\; \backslash Delta\; j\; \&=\; 0\; \backslash \backslash .2ex\backslash Delta\backslash left(vj^2\; +\; p\backslash right)\; \&=\; 0\; \backslash \backslash .2ex\backslash Delta\; h^t\; \&=\; 0\; \backslash end\{align\}\backslash right.,$
where $h^t$ is the specific total enthalpy.
These are the usually expressed in the convective variables:
:$\backslash left\backslash \{\backslash begin\{align\}\; \backslash Delta\; j\; \&=\; 0\; \backslash \backslash .2ex\backslash Delta\backslash left(\backslash frac\{u^2\}\{v\}\; +\; p\backslash right)\; \&=\; 0\; \backslash \backslash .2ex\backslash Delta\backslash left(e\; +\; \backslash frac\{1\}\{2\}u^2\; +\; pv\backslash right)\; \&=\; 0\; \backslash end\{align\}\backslash right.,$
where:
* $u$ is the flow speed
* $e$ is the specific internal energy.
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:
: $\backslash frac\{\backslash Delta\; p\}\{\backslash Delta\; v\}\; =\; -\; \backslash frac\; \{u\_0^2\}\{v\_0\}.$
Since the second term is a constant, the Rayleigh equation always describes a simple line in the pressure volume plane not depending of any equation of state, i.e. the Rayleigh line. By substitution in the Rankine–Hugoniot equations, that can be also made explicit as:
: $\backslash left\backslash \{\backslash begin\{align\}\; \backslash rho\; u\; \&=\; \backslash rho\_0\; u\_0\; \backslash \backslash .2ex\backslash rho\; u^2\; +\; p\; \&=\; \backslash rho\_0\; u\_0^2\; +\; p\_0\; \backslash \backslash .2exe\; +\; \backslash frac\{1\}\{2\}u^2\; +\; \backslash frac\{p\}\{\backslash rho\}\; \&=\; e\_0\; +\; \backslash frac\{1\}\{2\}u\_0^2\; +\; \backslash frac\{p\_0\}\{\backslash rho\_0\}\; \backslash end\{align\}\backslash right.\; .$
One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.
These are respectively:
: $\backslash left\backslash \{\backslash begin\{align\}\; u^2(v,\; p)\; \&=\; u\_0^2\; +\; \backslash left(p\; -\; p\_0\backslash right)\backslash left(v\_0\; +\; v\backslash right)\; \backslash \backslash .2exe(v,\; p)\; \&=\; e\_0\; +\; \backslash frac\{1\}\{2\}\; \backslash left(p\; +\; p\_0\backslash right)\backslash left(v\_0\; -\; v\backslash right)\; \backslash end\{align\}\backslash right.\; .$
The Hugoniot equation, coupled with the fundamental equation of state of the material:
:$e\; =\; e(v,p)$
describes in general in the pressure volume plane a curve passing by the conditions (v_{0}, p_{0}), i.e. the Hugoniot curve, whose shape strongly depends on the type of material considered.
It is also customary to define a ''Hugoniot function'':
: $\backslash mathfrak\; h\; (v,s)\; \backslash equiv\; e(v,s)\; -\; e\_0\; +\; \backslash frac\{1\}\{2\}\; \backslash left(p(v,s)\; +\; p\_0\backslash right)\backslash left(v\; -\; v\_0\backslash right)$
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.

Finite volume form

On the other hand, by integrating a generic conservation equation: :$\backslash frac\; \{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =\; \backslash mathbf\; s$ on a fixed volume V_{m}, and then basing on the divergence theorem, it becomes:
:$\backslash frac\; \{d\}\{dt\}\; \backslash int\_\{V\_m\}\; \backslash mathbf\; y\; dV\; +\; \backslash oint\_\{\backslash partial\; V\_m\}\; \backslash mathbf\; F\; \backslash cdot\; \backslash hat\; n\; ds\; =\; \backslash mathbf\; S\; .$
By integrating this equation also over a time interval:
: $\backslash int\_\{V\_m\}\; \backslash mathbf\; y(\backslash mathbf\; r,\; t\_\{n+1\})\; \backslash ,\; dV\; -\; \backslash int\_\{V\_m\}\; \backslash mathbf\; y(\backslash mathbf\; r,\; t\_n)\; \backslash ,\; dV+\; \backslash int\_\{t\_n\}^\{t\_\{n+1\; \backslash oint\_\{\backslash partial\; V\_m\}\; \backslash mathbf\; F\; \backslash cdot\; \backslash hat\; n\; \backslash ,\; ds\; \backslash ,\; dt\; =\; \backslash mathbf\; 0\; .$
Now by defining the node conserved quantity:
: $\backslash mathbf\; y\_\{m,n\}\; \backslash equiv\; \backslash frac\; 1\; \{V\_m\}\; \backslash int\_\{V\_m\}\; \backslash mathbf\; y(\backslash mathbf\; r,\; t\_n)\; \backslash ,\; dV\; ,$
we deduce the finite volume form:
: $\backslash mathbf\{y\}\_\{m,n+1\}=\backslash mathbf\{y\}\_\{m,n\}\; -\; \backslash frac\{1\}\{V\_m\}\; \backslash int\_\{t\_n\}^\{t\_\{n+1\; \backslash oint\_\{\backslash partial\; V\_m\}\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash hat\{n\}\backslash ,\; ds\; \backslash ,\; dt\; .$
In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:
: $\backslash left\backslash \{\backslash begin\{align\}\; \backslash mathbf\; u\_\{m,n\}\; \&=\; \backslash frac\{\backslash mathbf\; j\_\{m,n\{\backslash rho\_\{m,n\; \backslash \backslash .2exe\_\{m,n\}\; \&=\; \backslash frac\{E^t\_\{m,n\{\backslash rho\_\{m,n\; -\; \backslash frac\{1\}\{2\}u^2\_\{m,n\}\; \backslash \backslash .2ex\backslash end\{align\}\backslash right.\; .$
Then the explicit finite volume expressions of the original convective variables are:<
\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\.2ex \mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\.2ex \mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\.2ex\end{align}\right. .
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Constraints

It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.

Ideal polytropic gas

For an ideal polytropic gas the fundamental equation of state is: :$e(v,\; s)\; =\; e\_0\; e^\{(\backslash gamma-1)m\backslash left(s-s\_0\backslash right)\}\; \backslash left(\{v\_0\; \backslash over\; v\}\backslash right)^\{\backslash gamma-1\}$ where $e$ is the specific energy, $v$ is the specific volume, $s$ is the specific entropy, $m$ is the molecular mass, $\backslash gamma$ here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics. From this equation one can derive the equation for pressure by its thermodynamic definition: :$p(v,e)\; \backslash equiv\; -\; \{\backslash partial\; e\; \backslash over\; \backslash partial\; v\}\; =\; (\backslash gamma\; -\; 1)\; \backslash frac\; e\; v$ By inverting it one arrives to the mechanical equation of state: :$e(v,p)\; =\; \backslash frac\; \{pv\}\{\backslash gamma\; -\; 1\}$ Then for an ideal gas the compressible Euler equations can be simply expressed in the ''mechanical'' or ''primitive variables'' specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result: {Dt} &= v\nabla p + \mathbf{g} \\.2ex {Dp \over Dt} &= -\gamma p\nabla \cdot \mathbf{u} \end{align}\right. |cellpadding |border |border colour = #FF00FF |background colour = #ECFCF4 and in one-dimensional quasilinear form they results: :$\backslash frac\{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; t\}\; +\; \backslash mathbf\; A\; \backslash frac\{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; x\}\; =\; \{\backslash mathbf\; 0\}.$ where the conservative vector variable is: :$\{\backslash mathbf\; y\}=\backslash begin\{pmatrix\}v\backslash \backslash \; u\; \backslash \backslash p\; \backslash end\{pmatrix\}$ and the corresponding jacobian matrix is: : $\{\backslash mathbf\; A\}=\backslash begin\{pmatrix\}u\; \&\; -v\; \&\; 0\; \backslash \backslash \; 0\; \&\; u\; \&\; v\; \backslash \backslash \; 0\; \&\; \backslash gamma\; p\; \&\; u\; \backslash end\{pmatrix\}.$

** Steady flow in material coordinates **

In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the steady momentum Euler equation:
:$\backslash boldsymbol\{u\}\backslash cdot\backslash nabla\; \backslash boldsymbol\{u\}\; =\; -\; \backslash frac\{1\}\{\backslash rho\}\; \backslash nabla\; p,$
where $\backslash mathbf\; u$, $p$ and $\backslash rho$ denote the flow velocity, the pressure and the density, respectively.
Let $\backslash left\backslash \{\; \backslash mathbf\; e\_s,\; \backslash mathbf\; e\_n,\; \backslash mathbf\; e\_b\; \backslash right\backslash \}$ be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-hand side of the above equation, the convective derivative of velocity, can be described as follows:
:$\backslash begin\{align\}\; \backslash boldsymbol\{u\}\backslash cdot\backslash nabla\; \backslash boldsymbol\{u\}\; \backslash \backslash \; \&=\; u\backslash frac\{\backslash partial\}\{\backslash partial\; s\}(u\backslash boldsymbol\{e\}\_s)\; \&(\backslash boldsymbol\{u\}\; =\; u\; \backslash boldsymbol\{e\}\_s,~\; \{\backslash partial\; /\; \backslash partial\; s\}\; \backslash equiv\; \backslash boldsymbol\{e\}\_s\; \backslash cdot\; \backslash nabla)\; \backslash \backslash \; \&=\; u\backslash frac\{\backslash partial\; u\}\{\backslash partial\; s\}\backslash boldsymbol\{e\}\_s\; +\; \backslash frac\{u^2\}\{R\}\backslash boldsymbol\{e\}\_n\; \&(\backslash because~\; \backslash frac\{\backslash partial\backslash boldsymbol\{e\}\_s\}\{\backslash partial\; s\}\; =\; \backslash frac\{1\}\{R\}\backslash boldsymbol\{e\}\_n),\; \backslash end\{align\}$
where $R$ is the radius of curvature of the streamline.
Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:
:$\backslash begin\{cases\}\; \backslash displaystyle\; u\backslash frac\{\backslash partial\; u\}\{\backslash partial\; s\}\; =\; -\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; s\},\backslash \backslash \; \backslash displaystyle\; \{u^2\; \backslash over\; R\}\; =\; -\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; n\}\; \&(\{\backslash partial\; /\; \backslash partial\; n\}\backslash equiv\backslash boldsymbol\{e\}\_n\backslash cdot\backslash nabla),\backslash \backslash \; \backslash displaystyle\; 0\; =\; -\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; b\}\; \&(\{\backslash partial\; /\; \backslash partial\; b\}\backslash equiv\backslash boldsymbol\{e\}\_b\backslash cdot\backslash nabla).\; \backslash end\{cases\}$
For barotropic flow $(\backslash rho\; =\; \backslash rho(p))$, Bernoulli's equation is derived from the first equation:
:$\backslash frac\{\backslash partial\}\{\backslash partial\; s\}\backslash left(\backslash frac\{u^2\}\{2\}\; +\; \backslash int\backslash frac\{\backslash mathrm\{d\}p\}\{\backslash rho\}\backslash right)\; =\; 0.$
The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient.
The third equation expresses that pressure is constant along the binormal axis.

** Streamline curvature theorem **

Let $r$ be the distance from the center of curvature of the streamline, then the second equation is written as follows:
:$\backslash frac\{\backslash partial\; p\}\{\backslash partial\; r\}\; =\; \backslash rho\; \backslash frac\{u^2\}\{r\}~(>0),$
where $\{\backslash partial\; /\; \backslash partial\; r\}\; =\; -\{\backslash partial\; /\backslash partial\; n\}.$
This equation states:

Exact solutions

All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic. Solutions to the Euler equations with vorticity are: * parallel shear flows – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a Cartesian coordinate system $(x,y,z)$ the flow is for instance in the $x$-direction – with the only non-zero velocity component being $u\_x(y,z)$ only dependent on $y$ and $z$ and not on $x.$ * Arnold–Beltrami–Childress flow – an exact solution of the incompressible Euler equations. * Two solutions of the three-dimensional Euler equations with cylindrical symmetry have been presented by Gibbon, Moore and Stuart in 2003. These two solutions have infinite energy; they blow up everywhere in space in finite time.

See also

* Bernoulli's theorem * Kelvin's circulation theorem * Cauchy equations * Madelung equations * Froude number * Navier–Stokes equations * Burgers equation

References

Notes

Citations

Sources

* * * * * * * * * * * * * * * * * *

Further reading

* * * {{Leonhard Euler Category:Equations of physics Category:Equations of fluid dynamics Category:Leonhard Euler

History

The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in ''Mémoires de l'Académie des Sciences de Berlin'' in 1757 (in this article Euler actually published only the ''general'' form of the continuity equation and the momentum equation; the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816. During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector

Incompressible Euler equations with constant and uniform density

In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are: where: *$\backslash mathbf\; u$ is the flow velocity vector, with components in an ''N''-dimensional space $u\_1,\; u\_2,\; \backslash dots,\; u\_N$, *$=+\backslash mathbf\; u\backslash cdot\backslash nabla\backslash mathbf\; v$, for a generic function (or field) $\backslash mathbf\; v$ denotes its material derivative in time with respect to the advective field $\backslash mathbf\; u$ and *$\backslash nabla$ denotes the gradient with respect to space, *$\backslash cdot$ denotes the scalar product, *$\backslash nabla$ is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and *$\backslash nabla\; \backslash cdot\; \backslash mathbf\; u$ is the flow velocity divergence (second equation), *$w$ is the specific (with the sense of ''per unit mass'') thermodynamic work, the internal source term. * $\backslash mathbf$ represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on. The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become: :$\backslash left\backslash \; Euler\; equations\; in\; the\; Froude\; limit\; (no\; external\; field)\; are\; named\; free\; equations\; and\; are\; conservative.\; The\; limit\; of\; high\; Froude\; numbers\; (low\; external\; field)\; is\; thus\; notable\; and\; can\; be\; studied\; withperturbation\; theory.$

Conservation form

The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods. The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: :$\backslash frac+\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =,$ or simply in Einstein notation: :$\backslash frac+\; \backslash frac=\; 0\_i,$ where the conservation quantity $\backslash mathbf\; y$ in this case is a vector, and $\backslash mathbf\; F$ is a flux matrix. This can be simply proved. At last Euler equations can be recast into the particular equation:

Spatial dimensions

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., $x$ and $t$) 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.

Incompressible Euler equations

In convective form the incompressible Euler equations in case of density variable in space are: where the additional variables are: *$\backslash rho$ is the fluid mass density, *$p$ is the pressure, $p\; =\; \backslash rho\; w$. The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be: :$+\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; +\; \backslash rho\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u\; =\; 0$ but here the last term is identically zero for the incompressibility constraint.

Conservation form

The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: :$=\backslash begin\backslash rho\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; u\; \backslash \backslash 0\backslash end;\; \backslash qquad\; =\backslash begin\backslash rho\; \backslash mathbf\; u\backslash \backslash \backslash rho\; \backslash mathbf\; u\; \backslash otimes\; \backslash mathbf\; u\; +\; p\; \backslash mathbf\; I\backslash \backslash \backslash mathbf\; u\backslash end.$ Here $\backslash mathbf\; y$ has length $N+2$ and $\backslash mathbf\; F$ has size $(N+2)N$. In general (not only in the Froude limit) Euler equations are expressible as: :$\backslash frac\; \backslash begin\backslash rho\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; u\; \backslash \backslash 0\backslash end+\; \backslash nabla\; \backslash cdot\; \backslash begin\backslash rho\; \backslash mathbf\; u\backslash \backslash \backslash rho\; \backslash mathbf\; u\; \backslash otimes\; \backslash mathbf\; u\; +\; p\; \backslash mathbf\; I\backslash \backslash \; \backslash mathbf\; u\backslash end\; =\; \backslash begin0\; \backslash \backslash \; \backslash rho\; \backslash mathbf\; g\; \backslash \backslash \; 0\; \backslash end$

Conservation variables

The variables for the equations in conservation form are not yet optimised. In fact we could define: :$=\backslash begin\backslash rho\; \backslash \backslash \; \backslash mathbf\; j\; \backslash \backslash 0\backslash end;\; \backslash qquad\; =\backslash begin\; \backslash mathbf\; j\; \backslash \backslash \; \backslash frac\; \backslash rho\; \backslash ,\; \backslash mathbf\; j\; \backslash otimes\; \backslash mathbf\; j+\; p\; \backslash mathbf\; I\backslash \backslash \; \backslash frac\; \backslash mathbf\; j\; \backslash rho\; \backslash end.$ where: * $\backslash mathbf\; j\; =\; \backslash rho\; \backslash mathbf\; u$ is the momentum density, a conservation variable. where: * $\backslash mathbf\; f\; =\; \backslash rho\; \backslash mathbf\; g$ is the force density, a conservation variable.

Euler equations

In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation: &= -\frac + \mathbf \\.2ex &= -\frac\nabla \cdot \mathbf \end\right. |cellpadding |border |border colour = #FF0000 |background colour = #ECFCF4 where the additional variables here is: *$e$ is the specific internal energy (internal energy per unit mass). 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: :$\backslash left\backslash \}\; For\; a\; thermodynamic\; fluid,\; the\; compressible\; Euler\; equations\; are\; consequently\; best\; written\; as:\; \&=\; ve\_\backslash nabla\; v\; +\; ve\_\backslash nabla\; s\; +\; \backslash mathbf\; \backslash \backslash .2ex\&=\; 0\; \backslash end\backslash right.$ |cellpadding |border |border colour = #FFFF00 |background colour = #ECFCF4 where: * $v$ is the specific volume * $\backslash mathbf\; u$ is the flow velocity vector * $s$ is the specific entropy 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: :$+\; \backslash nabla\; \backslash cdot\; (\backslash rho\; s\; \backslash mathbf\; u)\; =\; 0$ 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: :$e\; =\; e(v,\; s)$ 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.

Conservation form

The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: :$=\; \backslash begin\; \backslash rho\; \backslash \backslash \; \backslash mathbf\; j\; \backslash \backslash \; E^t\; \backslash end;\; \backslash qquad\; =\; \backslash begin\; \backslash mathbf\; j\; \backslash \backslash \; \backslash frac\; 1\; \backslash rho\; \backslash mathbf\; j\; \backslash otimes\; \backslash mathbf\; j\; +\; p\; \backslash mathbf\; I\; \backslash \backslash \; \backslash left(E^t\; +\; p\backslash right)\; \backslash frac\backslash mathbf\; \backslash end.$ where: * $\backslash mathbf\; j\; =\; \backslash rho\; \backslash mathbf\; u$ is the momentum density, a conservation variable. * $E^t\; =\; \backslash rho\; e\; +\; \backslash frac\; \backslash rho\; u^2$ is the total energy density (total energy per unit volume). Here $\backslash mathbf\; y$ has length N + 2 and $\backslash mathbf\; F$ has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as: where: * $\backslash mathbf\; f\; =\; \backslash rho\; \backslash mathbf\; g$ is the force density, a conservation variable. 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: :$(\backslash rho\; s)\; +\; \backslash nabla\; \backslash cdot\; (\backslash rho\; s\; \backslash mathbf\; u)\; =\; 0$ Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as: \end = \begin0 \\ \mathbf \\ 0 \end |cellpadding |border |border colour = #FFFF00 |background colour = #ECFCF4 where: * $S\; =\; \backslash rho\; s$ is the entropy density, a thermodynamic conservation variable. Another possible form for the energy equation, being particularly useful for isobarics, is: :$\backslash frac\; +\; \backslash nabla\; \backslash cdot\; \backslash left(H^t\; \backslash mathbf\; u\backslash right)\; =\; \backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; f\; -\; \backslash frac$ where: *$H^t\; =\; E^t\; +\; p\; =\; \backslash rho\; e\; +\; p\; +\; \backslash frac\; \backslash rho\; u^2$ is the total enthalpy density.

Quasilinear form and characteristic equations

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 : :$\backslash frac\; +\; \backslash mathbf\; A\_i\; \backslash frac\; =\; .$ where $\backslash mathbf\; A\_i$ are called the flux Jacobians defined as the matrices: :$\backslash mathbf\; A\_i\; (\backslash mathbf\; y)=\backslash frac.$ Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians $\backslash mathbf\; A\_i$ are not functions of the state vector $\backslash mathbf\; y$, the equations reveals ''linear''.

Characteristic equations

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 $\backslash mathbf\; p\_i$ is the right eigenvector of the matrix $\backslash mathbf\; A$ corresponding to the eigenvalue $\backslash lambda\_i$, by building the projection matrix: :$\backslash mathbf\; =\; \backslash leftmathbf\_1,\; \backslash mathbf\_2,\; ...,\; \backslash mathbf\_n\backslash right/math>\; One\; can\; finally\; find\; the\; \text{'}\text{'}characteristic\; variables\text{'}\text{'}\; as:\; :$ \backslash mathbf=\; \backslash mathbf^\backslash mathbf,$Since\; A\; is\; constant,\; multiplying\; the\; original\; 1-D\; equation\; in\; flux-Jacobian\; form\; with\; P-1yields\; the\; characteristic\; equations:\; :$ \backslash frac\; +\; \backslash lambda\_j\; \backslash frac\; =\; 0\_i$The\; original\; equations\; have\; beendecoupledinto\; N+2\; characteristic\; equations\; each\; describing\; a\; simple\; wave,\; with\; the\; eigenvalues\; being\; the\; wave\; speeds.\; The\; variables\; \text{'}\text{'}w\text{'}\text{'}$

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) : :$\backslash left\backslash \{\backslash begin\{align\}\; \{\backslash partial\; v\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; v\; \backslash over\; \backslash partial\; x\}\; -\; v\; \{\backslash partial\; u\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash \backslash .2ex\{\backslash partial\; u\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; u\; \backslash over\; \backslash partial\; x\}\; -\; e\_\{vv\}\; v\; \{\backslash partial\; v\; \backslash over\; \backslash partial\; x\}\; -\; e\_\{vs\}v\; \{\backslash partial\; s\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash \backslash .2ex\{\backslash partial\; s\; \backslash over\; \backslash partial\; t\}\; +\; u\{\backslash partial\; s\; \backslash over\; \backslash partial\; x\}\; \&=\; 0\; \backslash end\{align\}\backslash right.$ If one defines the vector of variables: :$\backslash mathbf\{y\}\; =\; \backslash begin\{pmatrix\}v\; \backslash \backslash \; u\; \backslash \backslash \; s\backslash end\{pmatrix\}$ recalling that $v$ is the specific volume, $u$ the flow speed, $s$ the specific entropy, the corresponding jacobian matrix is: : $\{\backslash mathbf\; A\}=\backslash begin\{pmatrix\}u\; \&\; -v\; \&\; 0\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u\; \&\; -\; e\_\{vs\}\; v\; \backslash \backslash \; 0\; \&\; 0\; \&\; u\; \backslash end\{pmatrix\}.$ At first one must find the eigenvalues of this matrix by solving the characteristic equation: : $\backslash det(\backslash mathbf\; A(\backslash mathbf\; y)\; -\; \backslash lambda(\backslash mathbf\; y)\; \backslash mathbf\; I)\; =\; 0$ that is explicitly: : $\backslash det\backslash begin\{bmatrix\}u-\backslash lambda\; \&\; -v\; \&\; 0\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u-\backslash lambda\; \&\; -\; e\_\{vs\}\; v\; \backslash \backslash \; 0\; \&\; 0\; \&\; u-\backslash lambda\; \backslash end\{bmatrix\}=0$ This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements. : $(u-\backslash lambda)\; \backslash det\; \backslash begin\{bmatrix\}u-\backslash lambda\; \&\; -v\; \backslash \backslash \; -\; e\_\{vv\}\; v\; \&\; u\; -\backslash lambda\; \backslash end\{bmatrix\}=0$ Now by computing the determinant 2×2: : $(u\; -\; \backslash lambda)\backslash left((u\; -\; \backslash lambda)^2\; -\; e\_\{vv\}\; v^2\backslash right)\; =\; 0$ by defining the parameter: : $a(v,s)\; \backslash equiv\; v\; \backslash sqrt\; \{e\_\{vv$ or equivalently in mechanical variables, as: : $a(\backslash rho,p)\; \backslash equiv\; \backslash sqrt\; \{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}$ 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: : $\backslash begin\{pmatrix\}e\_\{vv\}\; \&\; e\_\{vs\}\; \backslash \backslash \; e\_\{vs\}\; \&\; e\_\{ss\}\; \backslash end\{pmatrix\}$ is defined positive. This statement corresponds to the two conditions: :$\backslash left\backslash \{\backslash begin\{align\}\; e\_\{vv\}\; \&>\; 0\; \backslash \backslash .2exe\_\{vv\}e\_\{ss\}\; -\; e\_\{vs\}^2\; \&>\; 0\; \backslash end\{align\}\backslash right.$ The first condition is the one ensuring the parameter ''a'' is defined real. The characteristic equation finally results: : $(u\; -\; \backslash lambda)\backslash left((u\; -\; \backslash lambda)^2\; -\; a^2\backslash right)\; =\; 0$ That has three real solutions: : $\backslash lambda\_1(v,u,s)\; =\; u-a(v,s)\; \backslash quad\; \backslash lambda\_2(u)=\; u,\; \backslash quad\; \backslash lambda\_3(v,u,s)\; =\; u+a(v,s)$ 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 λ

Compressibility and sound speed

Sound speed is defined as the wavespeed of an isentropic transformation: :$a\_s(\backslash rho,p)\; \backslash equiv\; \backslash sqrt\; \{\backslash left(\{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}\; \backslash right)\_s\}$ by the definition of the isoentropic compressibility: :$K\_s\; (\backslash rho,p)\; \backslash equiv\; \backslash frac\; 1\; \backslash rho\; \backslash left(\{\backslash partial\; p\; \backslash over\; \backslash partial\; \backslash rho\}\; \backslash right)\_s$ the soundspeed results always the square root of ratio between the isentropic compressibility and the density: :$a\_s\; \backslash equiv\; \backslash sqrt\; \{\backslash frac\; \{K\_s\}\; \backslash rho\}$

Ideal gas

The sound speed in an ideal gas depends only on its temperature: :$a\_s\; (T)\; =\; \backslash sqrt\; \{\backslash gamma\; \backslash frac\; T\; m\}$ Since the specific enthalpy in an ideal gas is proportional to its temperature: :$h\; =\; c\_p\; T\; =\; \backslash frac\; \{\backslash gamma\}\{\backslash gamma-1\}\; \backslash frac\; T\; m$ the sound speed in an ideal gas can also be made dependent only on its specific enthalpy: :$a\_s\; (h)\; =\; \backslash sqrt\; \{(\backslash gamma\; -1)\; h\}$

Bernoulli's theorem for steady inviscid flow

Bernoulli's theorem is a direct consequence of the Euler equations.

Incompressible case and Lamb's form

The vector calculus identity of the cross product of a curl holds: :$\backslash mathbf\{v\; \backslash \; \backslash times\; \}\; \backslash left(\; \backslash mathbf\{\; \backslash nabla\; \backslash times\; F\}\; \backslash right)\; =\; \backslash nabla\_F\; \backslash left(\; \backslash mathbf\{v\; \backslash cdot\; F\; \}\; \backslash right)\; -\; \backslash mathbf\{v\; \backslash cdot\; \backslash nabla\; \}\; \backslash mathbf\{\; F\}\; \backslash \; ,$ where the Feynman subscript notation $\backslash nabla\_F$ is used, which means the subscripted gradient operates only on the factor $\backslash mathbf\; F$. 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash mathbf\; u\; =\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; \backslash times\; \backslash mathbf\; u$ the Euler momentum equation in Lamb's form becomes: :$\backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; \backslash times\; \backslash mathbf\{u\}\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}\; =\; \backslash mathbf\{g\}\; =\; \backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\backslash nabla\backslash left(u^2\backslash right)\; -\; \backslash mathbf\{u\}\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}$ Now, basing on the other identity: :$\backslash nabla\; \backslash left(\; \backslash frac\; \{p\}\{\backslash rho\}\; \backslash right)\; =\; \backslash frac\; \{\backslash nabla\; p\}\{\backslash rho\}\; -\; \backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho$ the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows: :$\backslash nabla\; \backslash left(\backslash frac\{1\}\{2\}u^2\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; -\; \backslash mathbf\; g\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; -\; \backslash frac\{\backslash partial\; \backslash mathbf\; u\}\{\backslash partial\; t\}$ In fact, in case of an external conservative field, by defining its potential φ: :$\backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; -\; \backslash frac\{\backslash partial\; \backslash mathbf\; u\}\{\backslash partial\; t\}$ In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes: :$\backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash nabla\; \backslash rho\; +\; \backslash mathbf\; u\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)$ 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\backslash frac\{1\}\{2\}u^2\; +\; \backslash phi\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; =\; -\backslash frac\{p\}\{\backslash rho^2\}\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\backslash rho$ In the steady incompressible case the mass equation is simply: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; =\; 0$, 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; \backslash right)\; =\; 0$ The convenience of defining the total head for an inviscid liquid flow is now apparent: :$b\_l\; \backslash equiv\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; +\; \backslash frac\; p\; \backslash rho\; ,$ which may be simply written as: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; b\_l\; =\; 0$ 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.

Compressible case

In the most general steady (compressibile) case the mass equation in conservation form is: :$\backslash nabla\; \backslash cdot\; \backslash mathbf\; j\; =\; \backslash rho\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u\; +\; \backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash rho\; =\; 0$ . Therefore, the previous expression is rather :$\backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; \backslash left(\{\backslash frac\{1\}\{2u^2\; +\; \backslash phi\; +\; \backslash frac\{p\}\{\backslash rho\}\backslash right)\; =\; \backslash frac\{p\}\{\backslash rho\}\backslash nabla\; \backslash cdot\; \backslash mathbf\{u\}$ The right-hand side appears on the energy equation in convective form, which on the steady state reads: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; e\; =\; -\; \backslash frac\{p\}\{\backslash rho\}\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; u$ The energy equation therefore becomes: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; \backslash left(\; e\; +\; \backslash frac\; p\; \backslash rho\; +\; \backslash frac\; 1\; 2\; u^2\; +\; \backslash phi\; \backslash right)\; =\; 0,$ 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: :$h^t\; \backslash equiv\; e\; +\; \backslash frac\; p\; \backslash rho\; +\; \backslash frac\; 1\; 2\; u^2$ and the Bernoulli invariant for an inviscid gas flow is: :$b\_g\; \backslash equiv\; h^t\; +\; \backslash phi\; =\; b\_l\; +\; e\; ,$ which can be written as: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; b\_g\; =\; 0$ 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: :$\backslash mathbf\; u\; \backslash cdot\; \backslash nabla\; h^t\; \backslash sim\; 0$

Friedmann form and Crocco form

By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form: :$v\; \backslash nabla\; p\; =\; -T\; \backslash nabla\; s\; +\; \backslash nabla\; h$ in the convective form of Euler momentum equation, one arrives to: :$\backslash frac\{D\backslash mathbf\; u\}\{Dt\}=T\; \backslash nabla\backslash ,s-\backslash nabla\; \backslash ,h$ 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: :$\backslash frac\{\backslash partial\backslash mathbf\{u\{\backslash partial\; t\}\; +\; \backslash frac\{1\}\{2\}\; \backslash nabla\backslash left(u^2\backslash right)\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})\; \backslash times\; \backslash mathbf\{u\}\; +\; \backslash frac\{\backslash nabla\; p\}\{\backslash rho\}\; =\; \backslash mathbf\{g\}$ and by defining the specific total enthalpy: :$h^t\; =\; h\; +\; \backslash frac\{1\}\{2\}u^2$ one arrives to the Crocco–Vazsonyi form (Crocco, 1937) of the Euler momentum equation: :$\backslash frac\{\backslash partial\; \backslash mathbf\{\; u\{\backslash partial\; t\}\; +\; (\backslash nabla\; \backslash times\; \backslash mathbf\; u)\; \backslash times\; \backslash mathbf\; u\; -\; T\; \backslash nabla\; s\; +\; \backslash nabla\; h^t\; =\; \backslash mathbf\{g\}$ 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: :$\backslash left\backslash \{\backslash begin\{align\}\; \backslash mathbf\{u\}\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{u\}\; +\; T\backslash nabla\; s\; -\; \backslash nabla\; h^t\; \&=\; \backslash mathbf\{g\}\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; s\; \&=\; 0\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; h^t\; \&=\; 0\; \backslash end\{align\}\backslash right.$ Finally if the flow is also isothermal: :$T\; \backslash nabla\; s\; =\; \backslash nabla\; (T\; s)$ by defining the specific total Gibbs free energy: :$g^t\; \backslash equiv\; h^t\; +\; Ts$ the Crocco's form can be reduced to: :$\backslash left\backslash \{\backslash begin\{align\}\; \backslash mathbf\{u\}\; \backslash times\; \backslash nabla\; \backslash times\; \backslash mathbf\{u\}\; -\; \backslash nabla\; g^t\; \&=\; \backslash mathbf\{g\}\; \backslash \backslash \; \backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla\; g^t\; \&=\; 0\; \backslash end\{align\}\backslash right.$ From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.

Discontinuities

The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot equations. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See Navier–Stokes equations) Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur. To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the ''local'' forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are ''local variables'') of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.

Rankine–Hugoniot equations

Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain: :$\backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =\; \backslash mathbf\; 0$ where in general F is the flux matrix. By integrating this local equation over a fixed volume V

Finite volume form

On the other hand, by integrating a generic conservation equation: :$\backslash frac\; \{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; F\; =\; \backslash mathbf\; s$ on a fixed volume V

Constraints

It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.

Ideal polytropic gas

For an ideal polytropic gas the fundamental equation of state is: :$e(v,\; s)\; =\; e\_0\; e^\{(\backslash gamma-1)m\backslash left(s-s\_0\backslash right)\}\; \backslash left(\{v\_0\; \backslash over\; v\}\backslash right)^\{\backslash gamma-1\}$ where $e$ is the specific energy, $v$ is the specific volume, $s$ is the specific entropy, $m$ is the molecular mass, $\backslash gamma$ here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics. From this equation one can derive the equation for pressure by its thermodynamic definition: :$p(v,e)\; \backslash equiv\; -\; \{\backslash partial\; e\; \backslash over\; \backslash partial\; v\}\; =\; (\backslash gamma\; -\; 1)\; \backslash frac\; e\; v$ By inverting it one arrives to the mechanical equation of state: :$e(v,p)\; =\; \backslash frac\; \{pv\}\{\backslash gamma\; -\; 1\}$ Then for an ideal gas the compressible Euler equations can be simply expressed in the ''mechanical'' or ''primitive variables'' specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result: {Dt} &= v\nabla p + \mathbf{g} \\.2ex {Dp \over Dt} &= -\gamma p\nabla \cdot \mathbf{u} \end{align}\right. |cellpadding |border |border colour = #FF00FF |background colour = #ECFCF4 and in one-dimensional quasilinear form they results: :$\backslash frac\{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; t\}\; +\; \backslash mathbf\; A\; \backslash frac\{\backslash partial\; \backslash mathbf\; y\}\{\backslash partial\; x\}\; =\; \{\backslash mathbf\; 0\}.$ where the conservative vector variable is: :$\{\backslash mathbf\; y\}=\backslash begin\{pmatrix\}v\backslash \backslash \; u\; \backslash \backslash p\; \backslash end\{pmatrix\}$ and the corresponding jacobian matrix is: : $\{\backslash mathbf\; A\}=\backslash begin\{pmatrix\}u\; \&\; -v\; \&\; 0\; \backslash \backslash \; 0\; \&\; u\; \&\; v\; \backslash \backslash \; 0\; \&\; \backslash gamma\; p\; \&\; u\; \backslash end\{pmatrix\}.$

'' In a steady flow of an inviscid fluid without external forces, the center of curvature of the streamline lies in the direction of decreasing radial pressure. ''Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem". This "theorem" explains clearly why there are such low pressures in the centre of vortices, which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate lift forces.

Exact solutions

All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic. Solutions to the Euler equations with vorticity are: * parallel shear flows – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a Cartesian coordinate system $(x,y,z)$ the flow is for instance in the $x$-direction – with the only non-zero velocity component being $u\_x(y,z)$ only dependent on $y$ and $z$ and not on $x.$ * Arnold–Beltrami–Childress flow – an exact solution of the incompressible Euler equations. * Two solutions of the three-dimensional Euler equations with cylindrical symmetry have been presented by Gibbon, Moore and Stuart in 2003. These two solutions have infinite energy; they blow up everywhere in space in finite time.

See also

* Bernoulli's theorem * Kelvin's circulation theorem * Cauchy equations * Madelung equations * Froude number * Navier–Stokes equations * Burgers equation

References

Notes

Citations

Sources

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Further reading

* * * {{Leonhard Euler Category:Equations of physics Category:Equations of fluid dynamics Category:Leonhard Euler