Euler equations (fluid dynamics)
In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically, only the continuity and momentum equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – together as "the Euler equations".^{1}
Like the Navier–Stokes equations, the Euler equations are usually written in one of two forms: the "conservation form" and the "nonconservation form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The nonconservation form emphasizes changes to the state of a control volume as it moves with the fluid.
The Euler equations can be applied to compressible as well as to incompressible flow – using either an appropriate equation of state or assuming that the divergence of the flow velocity field is zero, respectively.
Contents
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;^{2} the energy conservation 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, thus it 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 PierreSimon Laplace in 1816.
During the second half of the 19th century, it was found that the equation related to the conservation 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.^{3}
Conservation and component form
In differential form, the equations are:
where
 ρ is the fluid mass density,
 u is the fluid velocity vector, with components u, v, and w,
 E = ρ e + ½ ρ ( u^{2} + v^{2} + w^{2} ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
 p is the pressure,
 denotes the tensor product, and
 0 is the zero vector.
These equations may be expressed in subscript notation. The second equation includes the divergence of a dyadic product, and may be clearer in subscript notation:
where the i and j subscripts label the three Cartesian components: ( x_{1}, x_{2}, x_{3} ) = ( x, y, z ) and ( u_{1}, u_{2}, u_{3} ) = ( u, v, w ). These equations may be more succinctly expressed using Einstein notation, in which matched indices imply a sum over those indices and and :
The above equations are expressed in conservation form, as this format emphasizes their physical origins (and is often the most convenient form for computational fluid dynamics simulations). By subtracting the velocity times the mass conservation term, the second equation (momentum conservation), can also be expressed as:
or, in vector notation:
which is Newton's second law of motion expressed in terms of the material derivative. Similarly, by subtracting the velocity times the above momentum conservation term, the third equation (energy conservation), can also be expressed as:
or
Conservation and vector form
In vector and conservation form, the Euler equations become:
where
This form makes it clear that f_{x}, f_{y} and f_{z} are fluxes.
The equations above thus represent conservation of mass, three components of momentum, and energy. There are thus five equations and six unknowns. Closing the system requires an equation of state; the most commonly used is the ideal gas law (i.e. p = ρ (γ−1) e, where ρ is the density, γ is the adiabatic index, and e the internal energy).
Note the odd form for the energy equation; see Rankine–Hugoniot equation. The extra terms involving p may be interpreted as the mechanical work done on a fluid element by its neighbor fluid elements. These terms sum to zero in an incompressible fluid.
The wellknown Bernoulli's equation can be derived by integrating Euler's equation along a streamline, under the assumption of constant density and a sufficiently stiff equation of state.
Nonconservation form with flux Jacobians
Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. From the original equations as given above in vector and conservation form, the equations are written in a nonconservation form as:
where A_{x}, A_{y} and A_{z} are called the flux Jacobians, which are matrices equal to:
Here, the flux Jacobians A_{x}, A_{y} and A_{z} are still functions of the state vector m, so this form of the Euler equations is nonlinear, just like the original equations. This nonconservation form is equivalent to the original Euler equations in conservation form, at least in regions where the state vector m varies smoothly.
Flux Jacobians for an ideal gas
The ideal gas law is used as the equation of state, to derive the full Jacobians in matrix form, as given below:^{4}

Flux Jacobians in matrix form for an ideal gas The xdirection flux Jacobian: The ydirection flux Jacobian:
The zdirection flux Jacobian:
Where .
The total enthalpy H is given by:
and the speed of sound a is given as:
Linearized form
The linearized Euler equations are obtained by linearization of the Euler equations in nonconservation form with flux Jacobians, around a state m = m_{0}, and are given by:
where A_{x,0}, A_{y,0} and A_{z,0} are the values of respectively A_{x}, A_{y} and A_{z} at some reference state m = m_{0}.
Uncoupled wave equations for the linearized onedimensional case
The Euler equations can be transformed into uncoupled wave equations if they are expressed in characteristic variables instead of conserved variables. As an example, the onedimensional (1D) Euler equations in linear fluxJacobian form is considered:
The matrix A_{x,0} is diagonalizable, which means it can be decomposed into:
Here r_{1}, r_{2}, r_{3} are the right eigenvectors of the matrix A_{x,0} corresponding with the eigenvalues λ_{1}, λ_{2} and λ_{3}.
Defining the characteristic variables as:
Since A_{x,0} is constant, multiplying the original 1D equation in fluxJacobian form with P^{−1} yields:
The equations have been essentially decoupled and turned into three wave equations, with the eigenvalues being the wave speeds. The variables w_{i} are called Riemann invariants or, for general hyperbolic systems, they are called characteristic variables.
Shock waves
The Euler equations are nonlinear hyperbolic equations and their general solutions are waves. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and socalled shock waves are formed; this is a nonlinear effect and represents the solution becoming multivalued. 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 shock conditions. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity. (See Navier–Stokes equations)
Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur.
The equations in one spatial dimension
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the onedimensional 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 (PDE's) degenerate into ordinary differential equations (ODE's). Numerical solutions of the Euler equations rely heavily on the method of characteristics.
Steady flow in streamline 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 momentum part of the Euler equations:^{5}
where v, p and ρ denote the velocity, the pressure and the density, respectively.
Let {e_{s}, e_{n}, e_{b} } 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 lefthanded side of the above equation, the substantial derivative of velocity, can be described as follows:
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:
For barotropic flow ( ρ=ρ(p) ), Bernoulli's equation is derived from the first equation:
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:
where
This equation states:
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 Englishlanguage scientific literature.^{6} Japanese fluiddynamicists call the relationship the "Streamline curvature theorem". ^{7}
This "theorem" explains clearly why there are such low pressures in the centre of vortices,^{6} which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate lift forces.^{6}
See also
Notes
 ^ Anderson, John D. (1995), Computational Fluid Dynamics, The Basics With Applications. ISBN 0071132104
 ^ E226  Principes generaux du mouvement des fluides
 ^ Christodoulou, Demetrios (October 2007). "The Euler Equations of Compressible Fluid Flow". Bulletin of the American Mathematical Society 44 (4): 581–602. doi:10.1090/S0273097907011810. Retrieved June 13, 2009.
 ^ See Toro (1999)
 ^ James A. Fay (June 1994). Introduction to Fluid Mechanics. MIT Press. ISBN 0262061651. see "4.5 Euler's Equation in Streamline Coordinates" pp.150pp.152 (http://books.google.com/books?id=XGVpue4954wC&pg=150)
 ^ ^{a} ^{b} ^{c} Babinsky, Holger (November 2003), "How do wings work?", Physics Education
 ^ 今井 功 (IMAI, Isao) (November 1973). 『流体力学(前編)』(Fluid Dynamics 1) (in Japanese). 裳華房 (Shoukabou). ISBN 4785323140.
Further reading
 Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.
 Thompson, Philip A. (1972). Compressible Fluid Flow. New York: McGrawHill. ISBN 0070644055.
 Toro, E.F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics. SpringerVerlag. ISBN 3540659668.
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