Euler equations¶

The Euler equations describe the motion of an inviscid fluid, one with no viscosity. They are the Navier–Stokes equations with the friction term removed, and they are the foundation of aerodynamics, gas dynamics and the theory of shock waves.

Published by Leonhard Euler in 1757, they were among the first partial differential equations written down for a physical system.

When viscosity can be ignored¶

Viscous effects scale as \(1/Re\), so the Euler equations are a good model when the Reynolds number is large, away from solid walls:

• airflow around a wing, outside the thin boundary layer
• shock waves and explosions
• sound waves
• large-scale atmospheric and ocean flows (with rotation added)

Near walls they fail: real fluids stick to surfaces (no-slip), and a thin boundary layer forms where viscosity always matters.

Incompressible form¶

For constant density \(\rho\):

Compare with Navier–Stokes: the viscous term \(\mu \nabla^2 \mathbf{u}\) is gone. That lowers the order of the equation from second to first in space, which changes which boundary conditions can be imposed (see below).

Compressible form¶

For gases at high speed, density, momentum and energy all change. Written in conservation form without body forces:

where \(E = \rho e + \tfrac{1}{2} \rho |\mathbf{u}|^2\) is the total energy per unit volume and \(e\) the internal energy per unit mass. For an ideal gas the system is closed by

with \(\gamma = c_p / c_v\) (\(\gamma \approx 1.4\) for air).

One-dimensional form¶

In 1D the whole system fits in one line, \(\partial_t \mathbf{U} + \partial_x \mathbf{F}(\mathbf{U}) = 0\), with

This is the standard test bed for numerical methods.

Waves and shocks¶

The compressible Euler equations are hyperbolic: information travels at finite speeds. In 1D those speeds are

the speed of sound. Two consequences:

• Mach number \(M = |\mathbf{u}| / c\) decides the character of the flow. For \(M < 1\) (subsonic) disturbances travel upstream; for \(M > 1\) (supersonic) they cannot.
• Shock waves: smooth solutions can steepen into discontinuities in finite time. Across a shock the equations hold in integral form, giving the Rankine–Hugoniot jump conditions.

For a moving shock with speed \(s\), they state that the flux jump equals \(s\) times the jump in conserved quantities:

Vorticity and circulation¶

Taking the curl of the incompressible equations (with a conservative body force) gives the vorticity equation, for \(\boldsymbol{\omega} = \nabla \times \mathbf{u}\):

The right-hand side is vortex stretching: a vortex tube that is stretched spins faster, like a figure skater pulling in their arms. In 2D this term vanishes, and vorticity is simply carried along with the flow.

Kelvin's circulation theorem: for an inviscid fluid with conservative body forces and density depending only on pressure, the circulation around any closed loop moving with the fluid is constant:

So a flow that starts irrotational stays irrotational, which is what makes potential flow theory possible.

Bernoulli's equation¶

Using the identity \((\mathbf{u} \cdot \nabla)\mathbf{u} = \nabla \left( \tfrac{1}{2} |\mathbf{u}|^2 \right) - \mathbf{u} \times \boldsymbol{\omega}\) and gravity \(\mathbf{f} = -\rho g \hat{\mathbf{z}}\), the steady incompressible Euler equations give

If the flow is also irrotational (\(\boldsymbol{\omega} = 0\)), the constant is the same everywhere. Writing \(\mathbf{u} = \nabla \phi\) then gives potential flow: \(\nabla^2 \phi = 0\), a linear equation that can be solved with classical methods.

Worked example: Torricelli's law¶

Water drains from a wide open tank through a small hole a depth \(h\) below the surface. Apply Bernoulli between the surface (point 1) and the jet (point 2). Both are at atmospheric pressure, and the surface moves very slowly, \(u_1 \approx 0\):

The water leaves as fast as if it had fallen freely from the surface. For \(h = 1\ \text{m}\), \(u_2 \approx 4.4\ \text{m/s}\).

Boundary conditions¶

Without viscosity, only the no-penetration condition can be imposed at a solid wall:

The fluid may slip freely along the surface. This is the root of the gap between ideal and real flows.

Relation to Navier–Stokes¶

Solving them numerically¶

Because solutions can contain shocks, numerical methods must handle discontinuities without spurious oscillations:

• Finite volume methods in conservation form, so that shocks move at the correct speed.
• Godunov's method and approximate Riemann solvers (Roe, HLL, HLLC) to compute fluxes between cells.
• High-resolution schemes (MUSCL, WENO) with limiters for sharp shocks and accurate smooth regions.
• Classic test case: the Sod shock tube, a 1D Riemann problem with a shock, a contact discontinuity and a rarefaction wave.

Open questions¶

Further reading¶

• G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
• J. D. Anderson, Modern Compressible Flow, McGraw-Hill.
• E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer.
• A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002.