Lecture 5

Question 1

Name all differences between linear and nonlinear transport processes.

Answer 1

1. Linear transport process:

df/dphi = k1 = const.
d2f/d2phi = 0.
The shape of the characteristics does not change.

2. non-linear transport processes:

df/dphi = lambda(phi).
d2f/d2phi not equal to 0
The shape of the characteristics changes.
- If dlambda/dx < 0, we will have convergent characteristics (compression characteristics)
- If dlambda/dx > 0, we will have divergent characteristics (expansion characteristics)

Question 2

What causes convergent and divergent characteristics?

Answer 2

It depends on the distribution of the characteristic velocity (dlambda/dx)

• If dlambda/dx < 0, we will have convergent characteristics (compression characteristics)
• If dlambda/dx > 0, we will have divergent characteristics (expansion characteristics)

Question 3

How does a shock form?

Answer 3

If you have convergent characteristics (dlambda/dx < 0), the slope of the characteristics reduce until the collapse in a single point. This is the shock formation.

This single point is called the breakdown point

Question 4

How can the shock speed Vs be computed?

Answer 4

We can apply Rankine-Hugoniot conditions for a moving control volume:
f(yL) - VsyL = f(yR) - VsyR. Where L is the inlet and R the outlet

Then, we can compute Vs , which is the velocity of the control volume and of the shock.

Question 5

How are characteristic velocities defined for systems of PDEs?

Answer 5

We need to compute the eigenvalues, and if they are real, they are supposed to be the characteristic velocities of the PDE.

Question 6

Give the compatibility conditions for 1-D time dependent Euler Equations.

Answer 6

1. Linear case:
2. dp - phocdu = 0 –> along x_dot = u - c
3. dp + phocdu = 0 –> along x_dot = u - c
4. ds = 0 –> along x_dot = u

u, c and pho have a hat. We have applied small perturbations, leaving only the average terms.

2. non-linear case, they are the massive equations.

They are in the slide 18. Know them by heart too

Question 7

Linearize the Euler Equations in their characteristic form

Answer 7

Done in the paper

Question 8

What’s the condition for the characteristic velocity for the system to be hyperbolic?

Answer 8

That the characteristic velocity (lambda_i) is real.