Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... Euler equations (fluid dynamics) From Wikipedia, the free encyclopedia 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.[1] 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 Potential flow around a wing. flow velocity is a solenoidal field, or using another appropriate energy This incompressible potential equation respectively (the simplest form for Euler equations being the flow satisfies the Euler conservation of the specific entropy). Historically, only the equations for the special case incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy of zero vorticity. equation – of the more general compressible equations together as "the Euler equations".[2] 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 usually "Lagrangian form", but this name is not self-explanatory and historically wrong, so it will be avoided) can also be put in the "conservation form" (also called usually "Eulerian form", but also this name is not self-explanatory and is historically wrong, so it will be avoided here). 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. Contents 1 History 2 Incompressible Euler equations with constant and uniform density 2.1 Properties 2.2 Nondimensionalisation 2.3 Conservation form 2.4 Spatial dimensions 3 Incompressible Euler equations 3.1 Conservation form 3.2 Conservation variables 4 Euler equations 4.1 Incompressible constraint 4.2 Enthalpy conservation 4.3 Thermodynamic systems 4.4 Conservation form 5 Quasilinear form and characteristic equations 5.1 Characteristic equations 5.2 Waves in 1D inviscid, nonconductive thermodynamic fluid 5.3 Compressibility and sound speed 5.3.1 Ideal gas 1 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... 6 Bernoulli's theorems for steady inviscid flow 6.1 Incompressible case and Lamb's form 6.2 Compressible case 6.3 Friedman form and Crocco form 7 Discontinuities 7.1 Rankine–Hugoniot equations 7.2 Finite volume form 8 Constraints 8.1 Ideal polytropic gas 8.2 Steady flow in material coordinates 8.2.1 Streamline curvature theorem 9 Exact solutions 10 See also 11 Notes 12 Further reading History The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Academie des Sciences de Berlin in 1757 (in this article Euler actually published only the general form of the continuity equation and the momentum equation;[3] 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.[4] 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:[5] Incompressible Euler equations with constant and uniform density (convective or Lagrangian form) where: is the flow velocity vector, with components in a N-dimensional space , denotes the material derivative in time, 2 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... denotes the scalar product, is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation), and is the convective operator, is the specific (with the sense of per unit mass) thermodynamic work, the internal source term. 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: In fact for a flow with uniform density the following identity holds: where is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not casual, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to: So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density being analyzed is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevancy. The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing scalar components, where is the physical dimension of the space of interest). In 3D for example and the and vectors are explicitly and . Flow velocity and pressure are the so-called physical variables.[1] These equations may be expressed in subscript notation: where the and subscripts label the N-dimensional space components. These equations may be more succinctly expressed using Einstein notation: 3 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... where the and subscripts label the N-dimensional space components, and ; is the Kroenecker delta. In 3D and the and vectors are explicitly and , and matched indices imply a sum over those indices and and . Properties Although Euler first presented these equations in 1755, many fundamental questions about them remain unanswered. In three space dimensions it is not even known whether solutions of the equations are defined for all time or if they form singularities.[6] Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: In the one dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers equation: This is a model equation giving many insights on Euler equations. Nondimensionalisation In order to make the equations dimensionless, a characteristic length , and a characteristic velocity , need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained: and of the field unit vector: Substitution of these inversed relations in Euler equations, defining the Froude number, yields 4 of 30 01/22/2017 11:09 AM Euler equations (fluid dynamics) - Wikipedia https://en.wikipedia.org/wiki/Euler_equations_(fluid_dyn... (omitting the * at apix): Incompressible Euler equations with constant and uniform density (nondimensional form) Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory. Conservation form The conservation form emphasizes the mathematical properties of Euler equations,