Euler equations
Encyclopedia
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".
Like the Navier-Stokes equations, the Euler equations are usually written in one of two forms: the "conservation
form" and the "non-conservation form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The non-conservation form emphasises 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.
, 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 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
.
where
The second equation includes the divergence
of a dyadic product, and may be clearer in subscript notation; for each j from 1 to 3 one has:
where the i and j subscripts label the three Cartesian components: ( x1 , x2 , x3 ) = ( x , y , z ) and ( u1 , u2 , u3 ) = ( u , v , w ).
Note that 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). The second equation, which represents momentum
conservation, can also be expressed in non-conservation form as:
but this form obscures the direct connection between the Euler equations and Newton's second law of motion.
where
This form makes it clear that fx, fy and fz are flux
es.
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 well-known 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.
es 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 non-conservation form as:
where Ax, Ay and Az are called the flux Jacobians, which are matrices
equal to:
Here, the flux Jacobians Ax, Ay and Az are still functions of the state vector m, so this form of the Euler equations is nonlinear, just like the original equations. This non-conservation form is equivalent to the original Euler equations in conservation form, at least in regions where the state vector m varies smoothly.
is used as the equation of state
, to derive the full Jacobians in matrix form, as given below:
The total enthalpy
H is given by:
and the speed of sound
a is given as:
where Ax,0 , Ay,0 and Az,0 are the values of respectively Ax, Ay and Az at some reference state m = m0.
equations if they are expressed in characteristic variables
instead of conserved variables. As an example, the one-dimensional (1-D) Euler equations in linear flux-Jacobian form is considered:
The matrix Ax,0 is diagonalizable
, which means it can be decomposed into:
Here r1, r2, r3 are the right eigenvectors of the matrix Ax,0 corresponding with the eigenvalues λ1, λ2 and λ3.
Defining the characteristic variables as:
Since Ax,0 is constant, multiplying the original 1-D equation in flux-Jacobian 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 wi are called Riemann invariants or, for general hyperbolic systems, they are called characteristic variables.
equations and their general solutions are waves
. 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 solution
s 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.
. This involves finding curves in plane of independent variables (i.e., x and t) along which partial differential equation
s (PDE's) degenerate into ordinary differential equation
s (ODE's). Numerical solutions
of the Euler equations rely heavily on the method of characteristics.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the Euler equations are a set of equations governing inviscid flow
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....
. They are named after Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
and heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
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".
Like the Navier-Stokes equations, the Euler equations are usually written in one of two forms: the "conservation
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
form" and the "non-conservation form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The non-conservation form emphasises changes to the state of a control volume as it moves with the fluid.
The Euler equations can be applied to compressible
Compressible flow
Compressible flow is the area of fluid mechanics that deals with fluids in which the fluid density varies significantly in response to a change in pressure. Compressibility effects are typically considered significant if the Mach number of the flow exceeds 0.3, or if the fluid undergoes very large...
as well as to incompressible flow
Incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
– using either an appropriate equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
or assuming that the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of the flow velocity
Flow velocity
In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
field is zero, respectively.
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. 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 conditionAdiabatic process
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...
, was supplied by Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
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
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
, and energy and momentum were likewise unified into a single concept, the energy-momentum vector
Four-momentum
In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...
.
Conservation and component form
In differential form, the equations are:where
- ρ is the fluid mass density,
- u is the fluid velocityVelocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
vector, with components u, v, and w, - E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energyEnergyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
per unit volumeVolumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
, with e being the internal energyInternal energyIn thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
per unit mass for the fluid, - p is the pressurePressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, and 
denotes the tensor productTensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
.
The second equation includes the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of a dyadic product, and may be clearer in subscript notation; for each j from 1 to 3 one has:
where the i and j subscripts label the three Cartesian components: ( x1 , x2 , x3 ) = ( x , y , z ) and ( u1 , u2 , u3 ) = ( u , v , w ).
Note that the above equations are expressed in conservation form
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
, as this format emphasizes their physical origins (and is often the most convenient form for computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
simulations). The second equation, which represents momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
conservation, can also be expressed in non-conservation form as:
but this form obscures the direct connection between the Euler equations and Newton's second law of motion.
Conservation and vector form
In vector and conservation form, the Euler equations become:where
This form makes it clear that fx, fy and fz are flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
es.
The equations above thus represent conservation of mass
Conservation of mass
The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...
, three components of momentum, and energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. There are thus five equations and six unknowns. Closing the system requires an equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
; the most commonly used is the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's 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 well-known Bernoulli's equation can be derived by integrating Euler's equation along a streamline
Streamlines, streaklines and pathlines
Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, streaklines and pathlines are field lines resulting from this vector field description of the flow...
, under the assumption of constant density and a sufficiently stiff equation of state.
Non-conservation form with flux Jacobians
Expanding the fluxFlux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
es can be an important part of constructing numerical solvers, for example by exploiting (approximate
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
) solutions to the Riemann problem
Riemann problem
A Riemann problem, named after Bernhard Riemann, consists of a conservation law together with piecewise constant data having a single discontinuity. The Riemann problem...
. From the original equations as given above in vector and conservation form, the equations are written in a non-conservation form as:
where Ax, Ay and Az are called the flux Jacobians, which are matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
equal to:
Here, the flux Jacobians Ax, Ay and Az are still functions of the state vector m, so this form of the Euler equations is nonlinear, just like the original equations. This non-conservation 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 lawIdeal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
is used as the equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
, to derive the full Jacobians in matrix form, as given below:
| Flux Jacobians in matrix form for an ideal gas |
|---|
The x-direction flux Jacobian: The y-direction flux Jacobian:  The z-direction flux Jacobian:  Where  . |
The total enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
H is given by:
and the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
a is given as:
Linearized form
The linearized Euler equations are obtained by linearization of the Euler equations in non-conservation form with flux Jacobians, around a state m = m0, and are given by:where Ax,0 , Ay,0 and Az,0 are the values of respectively Ax, Ay and Az at some reference state m = m0.
Uncoupled wave equations for the linearized one-dimensional case
The Euler equations can be transformed into uncoupled waveWave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
equations if they are expressed in characteristic variables
Method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...
instead of conserved variables. As an example, the one-dimensional (1-D) Euler equations in linear flux-Jacobian form is considered:
The matrix Ax,0 is diagonalizable
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
, which means it can be decomposed into:
Here r1, r2, r3 are the right eigenvectors of the matrix Ax,0 corresponding with the eigenvalues λ1, λ2 and λ3.
Defining the characteristic variables as:
Since Ax,0 is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields:
The equations have been essentially decoupled
Linear independence
In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
and turned into three wave equations, with the eigenvalues being the wave speeds. The variables wi are called Riemann invariants or, for general hyperbolic systems, they are called characteristic variables.
Shock waves
The Euler equations are nonlinear hyperbolicHyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
equations and their general solutions are waves
WAVES
The WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...
. Much like the familiar oceanic waves
Waves and shallow water
When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper,...
, waves described by the Euler Equations 'break'
Breaking wave
In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy...
and so-called shock waves
Shock Waves
Shock Waves, , is a 1977 horror movie written and directed by Ken Wiederhorn...
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 solution
Weak solution
In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
s 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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
. (See Navier–Stokes equations)
Shock propagation is studied – among many other fields – in aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
and rocket propulsion
Rocket
A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...
, 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 one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristicsMethod of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...
. This involves finding curves in plane of independent variables (i.e., x and t) along which partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (PDE's) degenerate into ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s (ODE's). Numerical solutions
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
of the Euler equations rely heavily on the method of characteristics.