# 4. Sod’s Test Problems: The Shock Tube Problem¶

This set of problems was introduced in the paper by Gary Sod in 1978 called “A Survey of Several Finite Difference Methods for Systems of Non-linear Hyperbolic Conservation Laws”

## 4.1. Assumptions¶

• 1D
• Infinitely long tube
• Inviscid fluid

## 4.2. Initial Conditions¶

• At t=0 the diaphragm is instantaneously removed (this is done experimentally using a a thin sheet of metal and a small explosion bursts the diaphragm)

## 4.3. Regions of Flow¶

• The bursting of the diaphragm causes a 1D unsteady flow consisting of a steadily moving shock - A Riemann Problem.
• 1 discontinuity is present
• The solution is self-similar with 5 regions
• Region 1 & 5 - left and right sides of initial states
• Region 2 - expansion or rarefaction wave (x-dependent state)
• Regions 3 & 4 - steady states independent of x within the region (uniform)

Contact line between 3 and 4 separates fluids of different entropy (but they have the same pressure and velocity) i.e. it’s an invisible line - e.g. two fluids one side with water and the other with dye - contact line is moving.

$p_3 = p_4$
$u_3 = u_4$

## 4.4. Sod’s Test Number 1¶

Unknowns:

• Pressure
• Velocity
• Speed of sound
• Density
• Entropy
• Mach Number

Can also use Euler Equations in Primitive Form with:

• Pressure
• Velocity
• Density

Vector notation for the Euler Equations with Primitive Variables, $$p, u, \rho$$

### 4.4.1. Initial Conditions¶

$\begin{split}\mathbf{V}(x,0) = \begin{cases} \mathbf{V}_L \quad x \lt 0 \\ \mathbf{V}_R \quad x \ge 0 \end{cases}\end{split}$
$\begin{split}\mathbf{V}_L = \begin{bmatrix} \rho_L \\ u_L \\ p_L \end{bmatrix} = \begin{bmatrix} 1 kg/m^3 \\ 0 m/s \\ 100 kN/m^2 \end{bmatrix}\end{split}$
$\begin{split}\mathbf{V}_R = \begin{bmatrix} \rho_R \\ u_R \\ p_R \end{bmatrix} = \begin{bmatrix} 0.125 kg/m^3 \\ 0 m/s \\ 10 kN/m^2 \end{bmatrix}\end{split}$
• Everything is quiet until you break the diaphragm (u=0)
• The pressure ratio is 10

### 4.4.2. Discretisation¶

• N = 50 points in [-10m, 10m]
• $$\Delta x$$ = 20m / 50 = 0.4m
• Initial CFL = 0.3
• Initial wave speed = 374.17m/s
• Timestep $$\Delta t$$ = 0.4(0.4/374.17) = 4.276 $$\times 10^{-4}$$
• $$\Delta t / \Delta x$$ = 1.069 $$\times 10^{-3}$$

Solution at t = 0.01s (in about 23 timesteps)

Now the problem is described, the numerical schemes can be applied.

## 4.5. Sod’s Test Number 2¶

Unknowns are same as Test Number 1

### 4.5.1. Initial Conditions¶

$\begin{split}\mathbf{V}_L = \begin{bmatrix} \rho_L \\ u_L \\ p_L \end{bmatrix} = \begin{bmatrix} 1 kg/m^3 \\ 0 m/s \\ 100 kN/m^2 \end{bmatrix}\end{split}$
$\begin{split}\mathbf{V}_R = \begin{bmatrix} \rho_R \\ u_R \\ p_R \end{bmatrix} = \begin{bmatrix} 0.01 kg/m^3 \\ 0 m/s \\ 1 kN/m^2 \end{bmatrix}\end{split}$

Pressure ratio is 100 - this test is harder

### 4.5.2. Discretisation¶

• N = 50 points in [-10m, 15m]
• $$\Delta x$$ = 25m / 50 = 0.5m
• Initial CFL = 0.3
• Initial wave speed = 374.17m/s
• Timestep $$\Delta t$$ = 0.3(0.5/374.17) = 4.01 $$\times 10^{-4}$$
• $$\Delta t / \Delta x$$ = 8.02 $$\times 10^{-4}$$

Solution at t = 0.01s (in about 25 timesteps)

Now the problem is described, the numerical schemes can be applied.

## 4.6. Test 1¶

### 4.6.1. Lax-Friedrichs¶

• Pressure has a jump due to shockwave
• Solution has numerical dissipation
• Odd-even decoupling is present (staircase pattern)
• Burgers Equation simulated all the important features of the Euler Equations

### 4.6.2. MacCormack¶

• Similar to inviscid Burgers
• Overshoot in pressure, speed of sound, density, entropy is bad
• Lax-Friedrichs is better than MacCormack

### 4.6.3. Richtmyer¶

• Less overshooting than MacCormack
• Undershoot in pressure is bad
• Overshoot in velocity is bad

## 4.7. Test 2¶

### 4.7.1. Lax-Friedrichs¶

• Diffusion
• Odd-even decoupling
• Speed of sound very diffused

### 4.7.2. MacCormack with Artificial Viscosity¶

• Smaller amplitude of oscillations even in Test 2
• Small number of points - is a hard test for numerical scheme (coarse mesh)
• Overshoot in velocity

### 4.7.3. Richtmyer with Artificial Viscosity¶

• Nice result - better than MacCormack