Introduction

In this exercise, you will investigate flow inside a convergent divergent nozzle and the unsteady wave motion in a shock tube using the Q1D solver available in CFLOW. In total, you will investigate four flow cases of which three are nozzle flows and one is the flow in a shock tube.

OBS Read chapters 5-7 in the course book before starting with the exercise!

When you have done the assignment, you should write a short report and hand in for assessment of this course element (instructions at the bottom of this page).

The following tasks should be solved both analytically and numerically. Note that in order to be able to apply the correct boundary conditions in the numerical simulations, you will need to **solve the problems analytically first!**

Task 1 - Quasi-1D simulation of the flow in a convergent-divergent nozzle

#### Task 1a - nozzle flow with internal shock

In this task you will simulate air flow through a convergent divergent nozzle. The exit and throat areas of the nozzle are 0.5 m\(^2\) and 0.25 m\(^2\), respectively. The nozzle inlet area can be set to 0.3 m\(^2\). The inlet reservoir pressure and temperature (total pressure and total temperature upstream of the nozzle inlet) are 1.0 atm and 288 K, respectively. In this first task, the exit static pressure (back pressure) is 0.6 atm. Setup the CFLOW solver using the recommended settings below and compare the results with your calculations. Is the shock location correct?

**Remember to export the solver state to a JSON-file when you are done with the task.**

#### Task 1b - choked subsonic nozzle flow

In this task you will simulate air flow through the same convergent divergent nozzle as in Task 1a but in this case the critical condition should be simulated (choked subsonic flow). Calculate the exit conditions corresponding to choked nozzle flow and set up the CFLOW according to the recommendations given below. Do the results look as expected? Are there any significant difference from the analytical solution?

**Remember to export the solver state to a JSON-file when you are done with the task.**

#### Task 1c - supercritical nozzle flow

In this task you will again simulate air flow through the same convergent divergent nozzle as in Task 1a but in this case the suoercritical condition should be simulated (perfectly matched supersonic flow). Calculate the exit conditions corresponding to supercritical nozzle flow and set up the CFLOW according to the recommendations given below. Compare the the results from the simulation with your analytical calculations.

**Remember to export the solver state to a JSON-file when you are done with the task.**

Task 2 - shock tube

In the second part of this assignment you will simulate the initial transient in a shock tube. Note that the solver settings are a bit different for this case since you will simulate moving shocks and expansion waves.

Initially (before the shock tube is started), the pressure ratio over the diaphragm is \(p_4/p_1=5.0\), the driver section pressure is \(p_4=500.0\ kPa\), and the temperature is \(T_4=T_1=300.0\ K\) in both the driver and the driven section of the tube. Both ends of the tube are solid walls. Set up the CFLOW solver according to the specifications given below.

Solve the problem analytically and compare with the results obtained with CFLOW.

You should calculate and compare

- The shock speed
- The head and tail speed of the expansion wave
- The velocity of the gas behind the shock
- The velocity of the reflected shock

One way of finding the shock, head, and tail velocities is produce several solutions with a few time steps (e.g. 250) between each solution (run the solver multiple times). By default, all solutions are saved and can be plotted in the same graphs for comparison purposes using the *flow data archive* functionality in CFLOW post (see example below). The speed can be estimated from the distance traveled from the center of the tube.

**Remember to export the solver state to a JSON-file when you are done with the task.**

Assignment Report

In order to complete the assignment, a report should be written and handed in for assessment. A suggested report structure is given below.

**When done, submit the report along with the exported JSON-files in Canvas**