Derivation of unsteady quasi-1D flow equations
Sint-Genesius-Rode, Mar 2018.
This page is in the "junk" directory! Make sure you check some better material as well, on
the main page
The quasi-1D isentropic expansion of a flow is a very well-studied problem, and its analytical solution can be
found on a large number of textbooks, such as thermodynamics, turbo machines, rockets design and fluid mechanics books.
However, the derivation of unsteady quasi-1D flows, although not really difficult, is somehow less common.
In this page you can find a derivation that I made on the blackboard.
It is very likely that I put some mistakes in there.
Right click on the images to make them larger.
Take the derivation with grano salis and tell me if you spot some errors! :-)
Let's start with the final result (we will explain and derive it in the following):
And let's start the derivation:
Step 0: prelude
Our goal is writing some differential equation along the axis of a channel,
such as a nozzle.
We will assume that quantities (density, velocity, pressure, temperature..) are constant on the cross
This assumption is the core of the quasi-1D models and to be satisfied it requires that the duct
changes relatively smoothly (abrupt changes of cross section would generate recirculations and the
velocity would reverse etc etc).
Note that the "uniformity over the cross section" assumption is not needed for the transverse velocity
component, since it does not appear in the equations (in fact, the only way for the flow to follow the
walls is having a non uniform transverse velocity.
We will start by writing down the quantities on a small volume "dV" that has a length (along the axis)
equal to "dx", and we define an averaged value of a generic quantiti "psi" on that element (you can think
at a finite volumes approach if you like):
Step 1: mass balance
Now that we defined that "average" over the (small) volume dV, we write down a balance of mass in it.
I just call our small volume "V" in the following.
The rate of change of mass in such volume is equal to the mass that enters/exits it by crossing its surface S:
As you can see, that average stuff disappeared by taking the limit for "dx" going to zero.
Step 2: momentum balance
Then we do the same for the momentum equation.
Notice that the side walls enter in the game now, through the pressure term:
Step 3: energy balance
And finally the energy equation, which is quite easy. Let us also introduce the ideal gas equation of state and the
relation among energy and temperature.
If you are interested in a script solving the steady nozzle equations, click
Again, if you spot any mistake, please let me know!