Euler equations (fluid dynamics)

In fluid dynamics, the Euler equations govern inviscid flow. They correspond to the Navier-Stokes equations with zero viscosity and heat conduction terms. They are usually written in the conservation form shown below to emphasize that they directly represent conservation of mass, momentum, and energy. The equations are named after Leonhard Euler.

The Euler equations can be applied to compressible as well as to incompressible flow — using either an appropriate equation of state or that the divergence of the flow velocity field is zero, repectively.

This page assumes that classical mechanics applies; see relativistic Euler equations for a discussion of compressible fluid flow when velocities approach the speed of light.

Euler equations in 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 ( u² + v² + w² )" is the total energy per unit volume, with "e" is the internal energy per unit mass for the fluid, and
*"p" is the pressure.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::$\{partial(\; ho\; u\_j)overpartial\; t\}+sum\_\{i=1\}^3\{partial(\; ho\; u\_i\; u\_j)overpartial\; x\_i\}+\{partial\; poverpartial\; x\_j\}=0,$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 .

Euler equations in 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 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.

Euler equations in non-conservation 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 non-conservation 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 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 law is used as the equation of state, to derive the full Jacobians in matrix form, as given below [See Toro (1999)] :

:

The total enthalpy "H" is given by:

:$H\; =E+frac\{p\}\{\; ho\},$

and the speed of sound "a" is given as:

:$a=sqrt\{frac\{gamma\; p\}\{\; ho\; =\; sqrt\{(gamma-1)left\; [H-frac\{1\}\{2\}left(u^2+v^2+w^2\; ight)\; ight]\; \}.$

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" = "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.

Transformation to uncoupled wave equations for the one-dimensional 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 one-dimensional (1-D) Euler equations in linear flux-Jacobian form is considered:

:

The matrix A_"x,0" is diagonalizable, which means it can be decomposed into:

:$mathbf\{A\}\_\{x,0\}\; =\; mathbf\{P\}\; mathbf\{Lambda\}\; mathbf\{P\}^\{-1\},$

:

:

Here "r₁", "r₂", "r₃" are the right eigenvectors of the matrix A_"x,0" corresponding with the eigenvalues "λ₁", "λ₂" and "λ₃".

Defining the "characteristic variables" as:

:$mathbf\{w\}=\; mathbf\{P\}^\{-1\}mathbf\{m\},$

Since A_"x,0" is constant, multiplying the original 1-D equation in flux-Jacobian form with P^-1 yields:

:$frac\{partial\; mathbf\{w\{partial\; t\}\; +\; mathbf\{Lambda\}\; frac\{partial\; mathbf\{w\{partial\; x\}\; =\; 0$

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".

hock 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 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 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 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., "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.

References

*cite book | first=G. K. | last=Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0521663962
*cite book | first=Philip A. | last=Thompson| year=1972 | title=Compressible Fluid Flow | publisher=McGraw-Hill | location=New York | isbn=0070644055
*cite book | first=E.F. | last=Toro | title=Riemann Solvers and Numerical Methods for Fluid Dynamics | publisher=Springer-Verlag | year=1999 | isbn=3-540-65966-8

Notes