Euler equations (fluid dynamics)

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.^[1]

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".^[2]

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).

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^[3] (although Euler had previously presented his work to the Berlin Academy in 1752).^[4] Prior work included contributions from the Bernoulli family as well as from Jean le Rond d'Alembert.^[5]

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.^[4]

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:^[6]

where:

• ${\displaystyle \mathbf {u} }$ is the flow velocity vector, with components in an N-dimensional space ${\displaystyle u_{1},u_{2},\dots ,u_{N}}$,
• ${\displaystyle {\frac {D{\boldsymbol {\Phi }}}{Dt}}={\frac {\partial {\boldsymbol {\Phi }}}{\partial t}}+\mathbf {v} \cdot \nabla {\boldsymbol {\Phi }}}$, for a generic function (or field) ${\displaystyle {\boldsymbol {\Phi }}}$ denotes its material derivative in time with respect to the advective field ${\displaystyle \mathbf {v} }$ and
• ${\displaystyle \nabla w}$ is the gradient of the specific (with the sense of per unit mass) thermodynamic work, the internal source term, and
• ${\displaystyle \nabla \cdot \mathbf {u} }$ is the flow velocity divergence.
• ${\displaystyle \mathbf {g} }$ 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: $${\displaystyle {\begin{aligned}{\partial \mathbf {u} \over \partial t}+(\mathbf {u} \cdot \nabla )\mathbf {u} &=-\nabla w+\mathbf {g} ,\\\nabla \cdot \mathbf {u} &=0.\end{aligned}}}$$

In fact for a flow with uniform density ${\displaystyle \rho _{0}}$ the following identity holds: $${\displaystyle \nabla w\equiv \nabla \left({\frac {p}{\rho _{0}}}\right)={\frac {1}{\rho _{0}}}\nabla p,}$$ where ${\displaystyle p}$ 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 causal, 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 nonuniform in space but constant in time, the continuity equation to be added to the above set would correspond to: $${\displaystyle {\frac {\partial \rho }{\partial t}}=0.}$$

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 discussed here is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevance.

The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing ${\displaystyle N}$ scalar components, where ${\displaystyle N}$ is the physical dimension of the space of interest). Flow velocity and pressure are the so-called physical variables.^[1]

In a coordinate system given by ${\displaystyle \left(x_{1},\dots ,x_{N}\right)}$ the velocity and external force vectors ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {g} }$ have components ${\displaystyle (u_{1},\dots ,u_{N})}$ and ${\displaystyle \left(g_{1},\dots ,g_{N}\right)}$, respectively. Then the equations may be expressed in subscript notation as:

$${\displaystyle {\begin{aligned}{\partial u_{i} \over \partial t}+\sum _{j=1}^{N}{\partial \left(u_{i}u_{j}+w\delta _{ij}\right) \over \partial x_{j}}&=g_{i},\\\sum _{i=1}^{N}{\partial u_{i} \over \partial x_{i}}&=0.\end{aligned}}}$$

where the ${\displaystyle i}$ and ${\displaystyle j}$ subscripts label the N-dimensional space components, and ${\displaystyle \delta _{ij}}$ is the Kroenecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent.

Although Euler first presented these equations in 1755, many fundamental questions or concepts about them remain unanswered.

In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.^[7]

Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: $${\displaystyle {\partial \over \partial t}\left({\frac {1}{2}}u^{2}\right)+\nabla \cdot \left(u^{2}\mathbf {u} +w\mathbf {u} \right)=0.}$$

In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers' equation: $${\displaystyle {\partial u \over \partial t}+u{\partial u \over \partial x}=0.}$$

This model equation gives many insights into Euler equations.

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In order to make the equations dimensionless, a characteristic length ${\displaystyle r_{0}}$, and a characteristic velocity ${\displaystyle u_{0}}$, 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: $${\displaystyle {\begin{aligned}u^{*}&\equiv {\frac {u}{u_{0}}},&r^{*}&\equiv {\frac {r}{r_{0}}},\\[5pt]t^{*}&\equiv {\frac {u_{0}}{r_{0}}}t,&p^{*}&\equiv {\frac {w}{u_{0}^{2}}},\\[5pt]\nabla ^{*}&\equiv r_{0}\nabla .\end{aligned}}}$$ and of the field unit vector: $${\displaystyle {\hat {\mathbf {g} }}\equiv {\frac {\mathbf {g} }{g}}.}$$

Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix):

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.

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.^[1]

The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: $${\displaystyle {\frac {\partial \mathbf {y} }{\partial t}}+\nabla \cdot \mathbf {F} ={\mathbf {0} },}$$ or simply in Einstein notation: $${\displaystyle {\frac {\partial y_{j}}{\partial t}}+{\frac {\partial f_{ij}}{\partial r_{i}}}=0_{i},}$$ where the conservation quantity ${\displaystyle \mathbf {y} }$ in this case is a vector, and ${\displaystyle \mathbf {F} }$ is a flux matrix. This can be simply proved.

At last Euler equations can be recast into the particular equation:

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., ${\displaystyle x}$ and ${\displaystyle t}$) 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.

In convective form the incompressible Euler equations in case of density variable in space are:^[6]

where the additional variables are:

• ${\displaystyle \rho }$ is the fluid mass density,
• ${\displaystyle p}$ is the pressure, ${\displaystyle p=\rho w}$.

The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be: $${\displaystyle {\partial \rho \over \partial t}+\mathbf {u} \cdot \nabla \rho +\rho \nabla \cdot \mathbf {u} =0,}$$

but here the last term is identically zero for the incompressibility constraint.

The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: $${\displaystyle \mathbf {y} ={\begin{pmatrix}\rho \\\rho \mathbf {u} \\0\end{pmatrix}};\qquad {\mathbf {F} }={\begin{pmatrix}\rho \mathbf {u} \\\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} \\\mathbf {u} \end{pmatrix}}.}$$

Here ${\displaystyle \mathbf {y} }$ has length ${\displaystyle N+2}$ and ${\displaystyle \mathbf {F} }$ has size ${\displaystyle (N+2)N}$.^[a] In general (not only in the Froude limit) Euler equations are expressible as: $${\displaystyle {\frac {\partial }{\partial t}}{\begin{pmatrix}\rho \\\rho \mathbf {u} \\0\end{pmatrix}}+\nabla \cdot {\begin{pmatrix}\rho \mathbf {u} \\\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} \\\mathbf {u} \end{pmatrix}}={\begin{pmatrix}0\\\rho \mathbf {g} \\0\end{pmatrix}}.}$$

The variables for the equations in conservation form are not yet optimised. In fact we could define: $${\displaystyle {\mathbf {y} }={\begin{pmatrix}\rho \\\mathbf {j} \\0\end{pmatrix}};\qquad {\mathbf {F} }={\begin{pmatrix}\mathbf {j} \\\mathbf {j} \otimes {\frac {1}{\rho }}\,\mathbf {j} +p\mathbf {I} \\{\frac {\mathbf {j} }{\rho }}\end{pmatrix}},}$$ where ${\displaystyle \mathbf {j} =\rho \mathbf {u} }$ is the momentum density, a conservation variable.

where ${\displaystyle \mathbf {f} =\rho \mathbf {g} }$ is the force density, a conservation variable.

In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation: