MAE2402 “Laboratory” Three
Numerical calculations of shock interaction.






Overview:
Due to the ongoing COVID pandemic, it has not been possible to either bring you into our gas-
dynamics laboratory, or send a demonstrator in to take the data necessary to run a high-quality
online laboratory. Instead of a true “laboratory”, we instead are asking you to complete what is
essentially a numerical assignment, considering shock-shock interactions in a double-ramp
compression, a geometry used in some scramjet inlets.
Motivation:
The assignment has been designed to force you to think about how you might solve some of the
compressible flow problems numerically. As part of this, you will need to write scripts to solve
various compressible flow equations. One of the outcomes you will have from this assignment is a
set of scripts that are generally useful for gas dynamics. These scripts may be very helpful to you on
the exam. This is a key part of why we have chosen to give you this assignment; we are hoping it will
also help you on the exam.

Good luck!



Introduction:
One of the many proposed designs for a scramjet inlet involves a double-ramp compression,
generating two oblique shocks to precompress the fluid before the shock-train in the combustor. An
example of such a configuration is presented in Figure 1.

Figure 1: Double-ramp scramjet inlet, from Nguyen, T., Behr, M., Reinartz, B., Hohn, O., & Gülhan, A.
(2013). Effects of sidewall compression and relaminarization in a scramjet inlet. Journal of Propulsion
and Power, 29(3), 628-638.
In a real (i.e. viscous) flow, an interaction between the shock structures and the boundary layer at
the start of the second ramp typically produces a flow separation, and the inviscid theory we are
applying in this unit cannot be used to produce realistic predictions (see an example of a simulated
viscous flow in Figure 2)


Figure 2: Numerical simulation of viscous flow in double-ramp scramjet inlet, from Nguyen, T., Behr,
M., Reinartz, B., Hohn, O., & Gülhan, A. (2013). Effects of sidewall compression and relaminarization
in a scramjet inlet. Journal of Propulsion and Power, 29(3), 628-638.

Problem:
For the purposes of this assignment, we will consider the purely inviscid interaction, even though we
acknowledge that the real flow will not look like this. A schematic representation of the problem is
presented in Figure 3. The angles 1 and 2 are dependent on the last digit of your student ID
number as per Table 1.
Last digit of ID 1 2
0 6 20
1 8 20
2 10 20
3 6 18
4 8 18
5 10 18
6 6 22
7 8 22
8 10 22
9 12 20
Table 1
The wavelike structures that result from this interaction should appear something like what is
sketched in Figure 3. Both compression corners will generate a shock, and these shocks will merge
into a single shock. Shocks are indicated in the schematic in red. The additional wave that forms
from the triple point may be a shock or an expansion wave, depending on the flow configuration.


Figure 3

You are tasked with calculating the properties of this flow as a function of Mach number. Assuming
that the incoming flow has the properties 1 = 300 and 1 = 100, produce plots of the
following variables, as a function of freestream Mach number :
a) Temperature difference across the line separating regions 3 and 4
b) Mach number in regions 3 and 4.

You will find that there are (many) Mach numbers for which you cannot produce a solution. There
are several possible reasons for this. In commenting on your results, explain why there is no possible
solution based on the configuration presented in Figure 3.

Even setting aside that you must vary Mach number, this is a problem that must be solved
iteratively. You should use Matlab or some equivalent software to produce your results.

Report:
Your report should contain the following:
1. A brief introduction to the problem, including a discussion of the various limitations of the
analysis to be undertaken.
2. All equations that you are using in your code to produce your results, including statements
of either what laws the equations are derived from, or what source you have used for the
equations.
3. A clear statement of the boundary conditions of the problem, i.e. what are the constraints
that determine the solution.
4. Plots of the above-mentioned variables as a function of Mach number in the range M = [1;5]
at a spacing of Δ = 0.1.
5. A discussion of any regions where a solution is not possible, explaining why it is not possible
(note that there is likely to be at least two different reasons).
6. An appendix including all code used to solve the problems and produce the plots..



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