In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.
The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is divergence-free. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including the energy equation – as "the compressible Euler equations".
The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations.
The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is useful from a numerical point of view).
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 (although Euler had previously presented his work to the Berlin Academy in 1752). Prior work included contributions from the Bernoulli family as well as from Jean le Rond d'Alembert.
The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow. An additional equation, which was 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 for compressible flows, and 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 the case of density constant in time and uniform in space are:
Incompressible Euler equations with constant and uniform density (convective or Lagrangian form)
where:
is the flow velocity vector field, with components in an N-dimensional space , is the material derivative of , describing the acceleration of the fluid, represents body accelerations (force per unit mass) acting on the fluid, for example acceleration due to gravity, electric fields, and so on, is the gradient of the specific pressure potential , is the divergence of the flow velocity vector field.
The first equation is the Euler momentum equation with uniform density. By expanding the material derivative, the equations become:
For a flow with uniform density , the specific pressure potential is defined by:
Consequently,
where is the mechanical pressure.
The second equation is the incompressible constraint, stating that the flow velocity is a solenoidal field. For an incompressible flow, the density of a fluid parcel remains constant along its motion. When the density is also uniform in space, this gives the divergence-free condition .
The continuity equation is
For constant density , this reduces to:
Thus, in the constant-density incompressible case, the continuity equation is equivalent to the incompressibility constraint and does not need to be introduced as an independent additional equation.
The equations above 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). Flow velocity and pressure are the principal physical variables.
In a coordinate system given by , the velocity and external force vectors and have components and , respectively. Then the equations may be expressed in subscript notation as:
where the and subscripts label the N-dimensional space components, and is the Kronecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent.
Properties
Although Euler first presented these equations in 1755, many fundamental questions concerning their mathematical properties remain open. In particular, the regularity and possible finite-time singularity formation of solutions of the three-dimensional incompressible Euler equations remain important questions in mathematical fluid dynamics.
In three spatial dimensions, there are special classes of solutions for which finite-time singularity formation has been established.
Smooth solutions of the free equations, in the sense of equations without a source term , satisfy conservation of kinetic energy. The local conservation law is:
Here . The conservation law follows from the momentum equation together with the divergence-free condition.
In the one-dimensional case without the source term and with no pressure gradient, the momentum equation becomes the inviscid Burgers' equation:
The inviscid Burgers equation is a fundamental model for nonlinear advection and conservation laws.
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 of order one. The following dimensionless variables are thus obtained:
For the direction of the external field, define the unit vector:
The Froude number is defined as:
Substitution of the dimensionless variables into the Euler equations and omission of the asterisks gives:
Incompressible Euler equations with constant and uniform density (nondimensional form)
The Froude number compares inertial effects with gravitational effects in fluid flow. A high Froude number corresponds to a regime in which the effect of gravity is relatively small.
The incompressible Euler equations with constant density can also be written in dimensional form as:
In the absence of an external field, , the Euler equations are homogeneous and can be written in conservation form.
Conservation form
The conservation form emphasizes the mathematical structure of the Euler equations and is particularly important in computational fluid dynamics. Conservative formulations are widely used in numerical methods for conservation laws and fluid dynamics.
For the free Euler equations, the momentum equation can be written as a conservation equation:
Together with the incompressibility constraint:
In Einstein notation, the same equations are:
where is the identity matrix and denotes the outer product.
Demonstration of the conservation form
For the divergence-free velocity field, the following identity holds:
In index notation:
This follows from the product rule and the incompressibility condition:
Similarly, because the identity tensor is constant in Cartesian coordinates:
In index notation:
Consequently, the momentum equation without an external field,
can be rewritten as:
This is the conservation, or Eulerian, form of the incompressible Euler momentum equation.
The system can be represented using a conservation variable and a flux matrix . The components of are:
and:
The components of the flux matrix corresponding to the momentum equations are:
The final row represents the incompressibility constraint:
Thus, in three dimensions, the conservation variables are:
The components of the corresponding flux matrix are:
For the equations with an external field, the momentum conservation equation takes the component form:
The additional constraint has zero source term:
Since and , this equation is precisely:
The conservation-law formulation of the Euler equations and its use in numerical methods are discussed by Toro.
Spatial dimensions
The Euler equations can be formulated in one, two, or three spatial dimensions. The one-dimensional equations are useful in simplified models and in certain problems involving approximately one-dimensional flow.
In one spatial dimension, the incompressible Euler equations reduce to:
The incompressibility constraint becomes:
For the one-dimensional free equation with no pressure gradient, the momentum equation reduces to the inviscid Burgers' equation:
In multiple spatial dimensions, the Euler equations contain one momentum equation for each spatial component of the velocity field. Numerical methods for the equations make extensive use of conservation laws, flux formulations, and, for suitable problems, 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: is the fluid mass density, is the pressure, .
The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be:
but here the last term is identically zero for the incompressibility constraint.
Conservation form
The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:
Here has length and has size . In general (not only in the Froude limit) Euler equations are expressible as:
Conservation variables The variables for the equations in conservation form are not yet optimised. In fact we could define:
where is the momentum density, a conservation variable.
where 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:
where the additional variable here is: 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 expands the material derivative, the equations above become:
Incompressible constraint (revisited)