Mass flow rate

To calculate the Mass flow rate (Mfr)  inside the engine, we have used a Fortran code provided by F.Ducros. With few modifications, this program is able to calculate the Masse Flow rate passing through a line (not a surface, because we are in 2D). It uses an interpolation with the values of the speed and the density at the mesh verteces close to this line.

The line we have used is plotted in red in the following graph. It is situated at the inlet of the engine.

Due to the configuration of the engine and to the line we have chose (a vertical), the only composant of the velocity that is needed in the calculation of the Mass Flow rate is the horizontal velocity (U).

Evolution with the incidence:

In a first approach, we can consider that the density is almost the same at the inside of the engine. As a consequence, the Mfr depends only on the velocity (U). If we note the incidence with an alpha, we have the following relation:

U=Uo * cos(alpha)

It will be interesting to compare the results given by the code with this "first order" approach.
The value of Uo is not the value of U at the inlet of the domain (far from the engine) but hte value of U at the inlet of the engine when the incidence is zero.

Remark: All the values of the Mass flow rate presented here are non-dimensionnal, ie it is divided by the incident total velocity and by the density far from the engine.

Subsonic case

The following graph shows a comparison in the subsonic case. The incidence angle are quite slow, but we needed to set the value of the censor to c=0.2 to be able to make the code converge. There is no shock in this case, but for incidence larger than 15 degrees, the gradient present needs to be taken into account.

The cosinus law is not a good aproach of the phenomenon according to this graph. The variations can be due to the fact that the flow is unsteady and that we compute it as a steady flow. More computations have to be done to know if we obtain always the same Mass flow for the same angle.

Supersonic case

The folowing graph shows a comparison for small inclination angle (between 0 and 12.5). The censor is equal to 0.5. With this value, we couldn't compute large incidence values, the code was bombing. That is why another graph is presented below. The value of the Mass flow seem to depend on the value of the censor, that is why the two sets of values are not presented together.

The difference are quite small, as we can see.

The following graph shows a comparison for high incidence angle, obtained with a censor equal to zero.

The evolution of the computed Mass flow in blue look like a cosinus, but with differences that are more important than in the forme case (low incident angle).
In conlusion , we can say that in the supersonic case, the comparison with a "first order" law is acceptable for small incidence angle, but is less precise for high angle.

Comparison of the censor effect. In  the following graph, we can see a diffenrence between the curves in blue and in red. The scale is quite streched, that is why it looks like a small jump at an order of 5% of the value.