Compressible flow · hyperbolic conservation laws

Shock Tube & Riemann Problem

Why does bursting the diaphragm in a shock tube send out a shock, a contact surface and a rarefaction fan? The one-dimensional Euler equations carry signals at three speeds: the flow speed minus the sound speed, the flow speed itself, and the flow speed plus the sound speed. The initial jump splits into one wave for each: a shock, a contact surface where density jumps but pressure and velocity match, and a rarefaction fan.

Set the left and right gas states separately, solve the one-dimensional Euler equations with HLL, Rusanov or HLLC finite-volume fluxes, and lay the exact Riemann solution over the result. The tube density, the x–t wave diagram, four profiles and the drift in conserved quantities update together.

Initial-state domainPositive non-vacuum states

Normalized time $0.0000$

Star pressure

p_*

$0.3031\,p_0$

Shock speed

S_s

$1.752\,a_0$

Density error

L_1

$0.00\,\%$

Mass drift $0.00\,\%$

Energy drift $0.00\,\%$

Wave familyrarefaction · contact · shock

Riemann workbench

Set the jump across the diaphragm

Exact Riemann overlayCompare HLL cells with the self-similar analytic solution.x–t characteristicsShow shock rays, contact speed, and rarefaction fans.Numerical cell markersReveal finite-volume sampling along every profile.Conservation drift panelReplace Mach profile with mass and energy drift.Left density

\rho_L/\rho_0

1.000

Common velocity shift

(u_L+u_R)/(2a_0)

0

Numerical flux comparison

F_{i+1/2}

\text{HLL}

Left pressure decade

\log_{10}(p_L/p_0)

1.00\,p_0

Right pressure decade

\log_{10}(p_R/p_0)

0.10\,p_0

Right density

\rho_R/\rho_0

0.125

Velocity difference

(u_R-u_L)/a_0

0.00\,a_0

Heat-capacity ratio

\gamma

1.40

Finite-volume cells

N

200

Courant number

C_{\mathrm{CFL}}

0.72

Evolution speed

s_t

1.00\times

Compare HLL, Rusanov and HLLC, which restores the contact wave, on the same initial states. Where a local star state is not positive, HLLC falls back to HLL. Density and pressure are set separately on each side, and the common shift and velocity difference set $u_L=\bar u-\Delta u/2,\quad u_R=\bar u+\Delta u/2$ . Expansion states that would produce a vacuum are rejected explicitly, and no exact solution is drawn for them.

The run stops before the fastest exact wave reaches an outflow boundary, so any conservation drift comes from the numerical scheme, not from flux leaving the tube.

This one-dimensional inviscid ideal-gas model is made dimensionless with $L,\rho_0,p_0,\quad a_0=\sqrt{p_0/\rho_0}$ ; this velocity unit is not the reference sound speed. You can compare HLL, Rusanov and HLLC. Initial states that would produce a vacuum are outside what the exact solver can handle, so the run pauses and explains why. Safety floors on density and pressure can break conservation, and this implementation does not prove that density and pressure stay positive everywhere.

Physics tutorial

Shock tubes: how one initial jump becomes three wave families

BackgroundA shock tube starts with two uniform gases separated by a diaphragm. When the diaphragm is removed, the single jump between them splits on its own into a shock, a contact discontinuity and a rarefaction fan. The shock compresses the gas, the contact separates the two gases, and the fan expands the gas smoothly.

Why it mattersIn 1978 Gary Sod made these initial states a standard test for compressible-flow codes. This Lab also uses the two-wave flux introduced by Harten, Lax and van Leer in 1983, and every numerical cell can be compared with the exact self-similar Riemann solution.

Start with the essentials

Focus question: Why does a single jump at the diaphragm produce a shock, a contact discontinuity and a rarefaction fan all at once?
One-sentence intuition: The one-dimensional Euler system has three characteristic speeds: $u-a$ , $u$ , and $u+a$ . The initial jump is shared out among these three families, which travel at different speeds, so one jump cannot stay a single wave.

Core mathematical model

One-dimensional Euler conservation law

\frac{\partial}{\partial t}\begin{bmatrix}\rho\\ \rho u\\ E\end{bmatrix}+\frac{\partial}{\partial x}\begin{bmatrix}\rho u\\ \rho u^2+p\\ u(E+p)\end{bmatrix}=0

The three rows express conservation of mass, momentum and total energy. The finite-volume method moves these quantities between neighboring cells only through fluxes at the cell faces.

Ideal-gas equation of state and sound speed

p=(\gamma-1)\!\left(E-\frac12\rho u^2\right),\qquad a=\sqrt{\frac{\gamma p}{\rho}}

The heat-capacity ratio links energy, pressure and sound speed. Changing it shifts the speeds of all three wave families.

HLL two-wave numerical flux

F_{\mathrm{HLL}}=\frac{S_R F_L-S_L F_R+S_LS_R(U_R-U_L)}{S_R-S_L}

When the left signal speed is negative and the right one positive, HLL uses a single averaged state between them. The flux stays conservative, but it smears contacts more than the exact solution does.

Stable time step (CFL condition)

\Delta t\le C_{\mathrm{CFL}}\frac{\Delta x}{\max_i\left(|u_i|+a_i\right)}

In one step the fastest wave must not cross a whole cell, so a finer grid automatically gets a shorter time step.

Common difficulties

A contact is not a weaker shock

Typical misconceptionDensity still jumps, so the middle line must be another small shock.

Better mental modelPressure and velocity are the same on both sides of a contact. Only the density and the kind of gas change, and the contact moves along with the gas in the star region.

A smeared shock does not mean a thick real shock

Typical misconceptionThe plotted front spans several cells, so a real inviscid shock must be that thick too.

Better mental modelIn the Euler equations a shock is a discontinuity. The width you see comes mainly from the numerical diffusion of the first-order HLL scheme and from the finite grid resolution.

Run the experiment

01
Reproduce the Sod benchmark
Choose Sod benchmark, keep the exact overlay visible, and run until the solver pauses automatically.
What to observe: A smooth rarefaction spreads to the left, the contact drifts with the flow in the middle, and the shock on the right runs ahead. The numerical curves follow the dashed exact solution closely.
02
Increase the pressure ratio
Switch to Strong shock, then raise the cell count and watch the density error.
What to observe: The shock speed and the star pressure rise sharply. More cells make the front narrower and the overall error smaller, but they also force smaller time steps.
03
Change which waves form
Compare Colliding flows with Symmetric expansion while the x–t diagram is visible.
What to observe: When the flows collide, both outer waves become shocks; when they move apart, both become rarefaction fans. The type of each wave depends on whether the star pressure is higher or lower than the initial pressure on that side.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.