Lecture 7 Flashcards
Very important thing to remember about Linear Waves?
Linear Waves are given by linear characteristic curves, which are related to constant characteristic velocities.
These linear waves can never intersect; as they have constant slopes. They are all parallel.
Which types of nonlinear time depended waves do exist?
- Smooth waves –> Expansion fan
- non-Smooth waves –> Shock waves and entropy waves.
Entropy waves are jumps in entropy and density (and probably in Temperature). But NO in pressure and velocity.
Are linear compression waves equal to shock waves?
Until some extent yes, because both of them make an increase in pressure and they change the velocity.
However, there’s no shock in linear compression because shock needs intersection of characteristics; and we know this does not happen in linear theory.
Why can rarefaction (=expansion) fans develop?
We have 2 parts of the expansion fan:
- The head of the expansion fan
- The tail of the expansion fan
We have very different characteristic velocities between these 2 points. Since this is a continues process (smooth changes - not a jump). Because of this, we can have the expansion fan.
Can the nonlinear compatibility relations used to compute shocks?
No
They require the solution to be smooth. However, they can compute isentropic compressions (smooth compressions) while these compressions have not lead to the formation of a shock wave.
How does velocity change across a rarefaction fan?
Same as in the linear case.
It increases in the opposite direction as the expansion fan is advancing.
Is there a difference in maximum velocity between steady and unsteady flow?
Make the demonstration of it
What does that mean?
Yes, it can be a difference.
They differ by a factor of sqrt(5)
See the demonstration in the paper.
That means that unsteady flows can reach higher velocities for a short time than steady flows.
Chacarcteristic quantities when we have time-dependent velocities (non-linear waves)?
Are the characteristic velocities and quantities coupled?
Slide 7
Yes, they are coupled. Now, u depends on, and therefore the characteristic curve x_dot = u - c is no longer linear.
What happens with the speed of sound after a expansion wave?
It decreases; at T decreases.
What are the properties (Temperature, Pressure of Speed of Sound) that has smooth changes when applying non-linear theory?
Pressure and Temperature during a non-linear expansion wave
Diagram how how variables change (p,T,pho,c,s and u) within a nonlinear waves
Exercise 7, last page
