Euler Equations (Compressible)

The Euler equations are the governing equations for fluid motion neglecting viscosity. In the compressible case, density changes, so they take the form of a coupled system of conservation laws including the energy equation. This is the starting point for compressible CFD.

In one dimension, they can be written using the conservative variable vector $\mathbf{U}$ and flux $\mathbf{F}$ as follows.

Here,

Exactly. In three dimensions, the momentum equation has three components, resulting in a system of five equations.

To close the system, an equation of state is needed. For an ideal gas, we use

For air, the specific heat ratio is $\gamma = 1.4$.

That's an important concept. For the one-dimensional Euler equations, the condition for being hyperbolic is that all eigenvalues of the Jacobian $\mathbf{A} = \partial\mathbf{F}/\partial\mathbf{U}$ are real. The eigenvalues are

where $a = \sqrt{\gamma p/\rho}$ is the speed of sound. These are called characteristic speeds and represent the speeds at which information propagates.

$\lambda_1$ and $\lambda_3$ correspond to forward and backward acoustic waves, and $\lambda_2$ corresponds to an entropy wave (contact discontinuity). In supersonic flow ($|u| > a$), all eigenvalues have the same sign, meaning information propagates in only one direction. This directly affects how boundary conditions are set in CFD.

At a subsonic inlet, information like pressure propagates upstream, so one physical quantity is specified from the outside and the rest are extrapolated from the interior. At a supersonic inlet, all characteristics are incoming, so all variables must be specified. If this principle is not followed, unphysical reflected waves will occur.

Exactly. Solutions to the Euler equations include discontinuous solutions like shock waves and contact discontinuities. The techniques to capture these correctly numerically are the core technology of compressible CFD.

In Godunov's method, the numerical flux is obtained by solving a local Riemann problem at the cell interface.

Here, $\mathbf{U}^*$ is the solution to the Riemann problem, and $\mathbf{U}_L, \mathbf{U}_R$ are the left and right states at the cell interface. Exact Riemann solvers are computationally expensive, so approximate Riemann solvers are widely used.

Roe computes Roe-averaged quantities from the left and right states to linearize the Jacobian. The Roe-averaged density is defined as $\hat{\rho} = \sqrt{\rho_L \rho_R}$. The numerical flux is

The second term is the upwind term that adds numerical dissipation.

With first-order Godunov schemes, shock waves can spread over dozens of cells. For high-order accuracy, the MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) method is used.

Here, $\phi(r)$ is the slope limiter function. Without a limiter, Gibbs oscillations occur near shock waves. Representative limiters include minmod, van Leer, and superbee.

It changes them quite a bit. minmod is the most diffusive but stable, superbee is sharp but tends to make expansion waves stair-stepped. Practically, van Leer or van Albada offer a good balance. WENO (Weighted Essentially Non-Oscillatory) schemes offer fifth-order accuracy and excellent shock capturing, but are computationally expensive.

For explicit methods, TVD Runge-Kutta methods (Shu-Osher 3-stage 3rd-order) are standard. The CFL condition is

with CFL $\leq 1$ being the stability condition. For implicit methods, the LU-SGS (Lower-Upper Symmetric Gauss-Seidel) method is efficient, allowing larger CFL numbers, which speeds up convergence for steady-state calculations.