Euler equations (fluid dynamics)

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

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 "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.

1 History
2 Conservation and component form
3 Conservation and vector form
4 Non-conservation form with flux Jacobians
5 Shock waves
6 The equations in one spatial dimension
7 See also
8 Notes
9 Further reading

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 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 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.^[2]

Conservation and component form

In differential form, the equations are:

$\begin{align} &{\partial\rho\over\partial t}%2B \nabla\cdot(\rho\bold u)=0\\[1.2ex] &{\partial\rho{\bold u}\over\partial t}%2B \nabla\cdot(\bold u\otimes(\rho \bold u))%2B\nabla p=0\\[1.2ex] &{\partial E\over\partial t}%2B \nabla\cdot(\bold u(E%2Bp))=0, \end{align}$

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 being the internal energy per unit mass for the fluid,
p is the pressure, and
$\otimes$ denotes the tensor product.

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:

${\partial(\rho u_j)\over\partial t}%2B \sum_{i=1}^3 {\partial(\rho u_i u_j)\over\partial x_i}%2B {\partial p\over\partial x_j} =0,$

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:

$\rho\left( \frac{\partial}{\partial t}%2B{\bold u}\cdot\nabla \right){\bold u}%2B\nabla p=0$

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:

$\frac{\partial \bold m}{\partial t}%2B \frac{\partial \bold f_x}{\partial x}%2B \frac{\partial \bold f_y}{\partial y}%2B \frac{\partial \bold f_z}{\partial z}=0,$

where

${\bold m}=\begin{pmatrix}\rho \\ \rho u \\ \rho v \\ \rho w \\E\end{pmatrix};$

${\bold f_x}=\begin{pmatrix}\rho u\\p%2B\rho u^2\\ \rho uv \\ \rho uw\\u(E%2Bp)\end{pmatrix};\qquad {\bold f_y}=\begin{pmatrix}\rho v\\ \rho uv \\p%2B\rho v^2\\ \rho vw \\v(E%2Bp)\end{pmatrix};\qquad {\bold f_z}=\begin{pmatrix}\rho w\\ \rho uw \\ \rho vw \\p%2B\rho w^2\\w(E%2Bp)\end{pmatrix}.$

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.

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:

$\frac{\partial \bold m}{\partial t} %2B \bold A_x \frac{\partial \bold m}{\partial x} %2B \bold A_y \frac{\partial \bold m}{\partial y} %2B \bold A_z \frac{\partial \bold m}{\partial z} = 0.$

where A_x, A_y and A_z are called the flux Jacobians, which are matrices equal to:

$\bold A_x=\frac{\partial \bold f_x(\bold s)}{\partial \bold s}, \qquad \bold A_y=\frac{\partial \bold f_y(\bold s)}{\partial \bold s} \qquad \text{and} \qquad \bold A_z=\frac{\partial \bold f_z(\bold s)}{\partial \bold s}.$

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^[3]:

Flux Jacobians in matrix form for an ideal gas

The x-direction flux Jacobian:

$\bold A_x= \left[ \begin{array}{c c c c c} 0 & 1 & 0 & 0 & 0 \\ \hat{\gamma}H-u^2-a^2 & (3-\gamma)u & -\hat{\gamma}v & -\hat{\gamma}w & \hat{\gamma} \\ -uv & v & u & 0 & 0 \\ -uw & w & 0 & u & 0 \\ u[(\gamma-2)H-a^2] & H-\hat{\gamma}u^2 & -\hat{\gamma}uv & -\hat{\gamma}uw & \gamma u \end{array} \right].$

The y-direction flux Jacobian:

$\bold A_y= \left[ \begin{array}{c c c c c} 0 & 0 & 1 & 0 & 0 \\ -vu & v & u & 0 & 0 \\ \hat{\gamma}H-v^2-a^2 & -\hat{\gamma}u & (3-\gamma)v & -\hat{\gamma}w & \hat{\gamma} \\ -vw & 0 & w & v & 0 \\ v[(\gamma-2)H-a^2] & -\hat{\gamma}uv & H-\hat{\gamma}v^2 & -\hat{\gamma}vw & \gamma v \end{array} \right].$

The z-direction flux Jacobian:

$\bold A_z= \left[ \begin{array}{c c c c c} 0 & 0 & 0 & 1 & 0 \\ -uw & w & 0 & u & 0 \\ -vw & 0 & w & v & 0 \\ \hat{\gamma}H-w^2-a^2 & -\hat{\gamma}u & -\hat{\gamma}v & (3-\gamma)w& \hat{\gamma} \\ w[(\gamma-2)H-a^2] & -\hat{\gamma}uw & -\hat{\gamma}vw & H-\hat{\gamma}w^2 & \gamma w \end{array} \right].$

Where $\hat{\gamma}=\gamma-1$ .

The total enthalpy H is given by:

$H = \frac{E}{\rho} %2B \frac{p}{\rho},$

and the speed of sound a is given as:

$a=\sqrt{\frac{\gamma p}{\rho}} = \sqrt{(\gamma-1)\left[H-\frac{1}{2}\left(u^2%2Bv^2%2Bw^2\right)\right]}.$

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:

$\frac{\partial \bold m}{\partial t} %2B \bold A_{x,0} \frac{\partial \bold m}{\partial x} %2B \bold A_{y,0} \frac{\partial \bold m}{\partial y} %2B \bold A_{z,0} \frac{\partial \bold m}{\partial z} = 0,$

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 = m₀.

Uncoupled wave equations for the linearized 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:

$\frac{\partial \bold m}{\partial t} %2B \bold A_{x,0} \frac{\partial \bold m}{\partial x} =0.$

The matrix A_x,0 is diagonalizable, which means it can be decomposed into:

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

$\mathbf{P}= \left[\bold r_1, \bold r_2, \bold r_3\right] =\left[ \begin{array}{c c c} 1 & 1 & 1 \\ u-a & u & u%2Ba \\ H-u a & \frac{1}{2} u^2 & H%2Bu a \\ \end{array} \right],$

$\mathbf{\Lambda} = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \\ \end{bmatrix} = \begin{bmatrix} u-a & 0 & 0 \\ 0 & u & 0 \\ 0 & 0 & u%2Ba \\ \end{bmatrix}.$

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:

$\frac{\partial \mathbf{w}}{\partial t} %2B \mathbf{\Lambda} \frac{\partial \mathbf{w}}{\partial x} = 0$

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.

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 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.

Notes

^ Anderson, John D. (1995), Computational Fluid Dynamics, The Basics With Applications. ISBN 0-07-113210-4
^ Christodoulou, Demetrios (October 2007). "The Euler Equations of Compressible Fluid Flow". Bulletin of the American Mathematical Society 44 (4): 581–602. doi:10.1090/S0273-0979-07-01181-0. http://www.ams.org/bull/2007-44-04/S0273-0979-07-01181-0/S0273-0979-07-01181-0.pdf. Retrieved June 13, 2009.
^ See Toro (1999)