How to design a gradual expansion nozzle by the method of characteristics ?
 

Hi,

As the title says, I am looking to design a gradual expansion nozzle by the method of characteristics. I found the following sketch in the Anderson's book where the mathematical expressions giving such a nozzle is missing. The explanation about this image in the book is only about the presence of non-simple regions (please look at Anderson_GradualExpansion).

As it can be seen, in the expansion section, a series of waves are reflected several times by the wall before reaching the straightening section. However, the application of Nasa shows that there is no reflective waves ( please look at Nasa_GradualExpansion).

I am puzzled. I assume that the second image is a particular case of the first one: for a given situation (that I don't know how it's characterized), the first picture is simplified to the second one. Could you suggest me any textbook where the thorough design of a gradual expansion nozzle by MOC is explained ? Most of the books explain the minimum expansion length by MOC.


Best regards,
Mary

Have you already read the books of Zucrow?

No. Does he answer to my question in his book ? I read the one of Anderson but as I told you, he does not provide any detailed answer to my question.

I have Zucrow's book. I just had a look but I don't see any explanation about gradual expansion nozzles.

Have you also vol 2 ?

I had the vol. 1. Now, I found vol.2. Could you please tell me where exactly in his book my question is treated ? Out of curiosity, could you tell me under which kind of conditions, the figure of Anderson is simplified to the one of Nasa ? In the code that I should write, I would like to check/implement this transition.

I don't see conceptual differences, Anderson provides a sketch, in the NASA figure the reflection is reported until the last point on the wall.

I mean the expansion part. The two images are totally different ! in Nasa, you dont have any reflection of expansion waves in the expansion part. But in Anderson, the expansion waves are several times reflected before they reach the straightening part of the nozzle. In attached figure, what is the number of characteristic lines ? 3 or 9 ? They are three lines which are reflected 6 times (blue, green, purple). I think that the desired number of characteristic lines has an impact on the accuracy and the number of reflection. In Nasa, the characteristic lines are reflected only two times !

Could you provide the link of the NASA solution so that I can see better? It seems they simply do no longer plot the reflected characteristic lines that go out of the solid domain.
However, I think that you can find all is required in Chap 16 and 17 of Vol.2 in Zucrow. The former chapter is for omoentropic (you have the exact Riemann invariant) the latter for isoentropic flows.

It is an applet by Nasa which is called MOC. You can download the app here: https://www.grc.nasa.gov/WWW/K-12/airplane/mocnoz.html

Have you tried to plot the flow variables that are computed (Flow-Geometry option)?

Yes. But it does not help me to get an answer to my question. In Nasa geometry, you take an arc of a circle (theta). You decide for the number of characteristic that you want (n). You space points (from which characteristic lines are issued) by x*theta/n where (x=1,..n) and you go for conventional MOC. In picture that I sent you, there should be some conditions that limits you to freely choose your parameters. You can not decide for a given number of characteristic lines. You start, you analyze if there is reflection or not. If the reflection point is before the end of the circle are, you add characteristic lines. So, it seems we are not so free to choose any desired number of characteristics (in previous image, we can only take 3 and not more).There should be a link between the number of characteristic lines, the expansion length, etc. What I wanted to understand was the underlying phenomena or an algorithm through which these mutual interactions could be understood.

For omoentropic flow, the Riemann invariants allows to determine an analytical solution in the whole domain. There are two families of characteristic curves that exist in any point of the domain. The fact that you plot only a finite number of curves is only for graphics reason.

This flow is exactly the same you can analyse for the omoentropic expansion in a 1D unsteady flow where a piston is moving (you have simple waves) as shown in Vol1 of Zucrow. The solution is analytical and determined everywhere but for simple waves only a family is relevant.
I think that the representation in the NASA plot is limited by some graphic constraint that show only a finite number of lines representing only the gradual expansion fan. For example here you can see in this picture the two system of centred waves

https://www.grc.nasa.gov/WWW/K-12/Un...ges/mocnzl.jpg


That does not mean that there are no other characteristic lines but they are not represented since only a family is relevant.

But if the applet determines the solution in the whole domain that means you have the characteristic lines in the whole domain, including the reflected waves that are not shown.

Thanks Filippo for your explanation. Let's reformulate my question: I am going to write down the code by myself to have a 2D gradual expansion nozzle by MOC. Do you mean that if I code to get something like what is illustrated by the Nasa image, the CFD results of the latter it would be like the case where the reflection of waves by the center line is coded ? Am I going to have the same results shown in flow-geometry (you said that the reflection of waves might have been taken into account in the code but they are not shown) ? I have doubts. The reason of creating this thread was to know how to code a 2D gradual-expansion nozzle.

As you can understand from the whole picture, the reflected waves are nothing but the characteristic lines of the other family coming from the bottom wall.
The whole method is analytic.
I suggest to read the vol.1 of zucrow, the sections where the expansion waves from a piston moving in a tube is described. You can understand the steady problem in x,y plane from the unsteady problem in x,t plane.
Note that the theory is valid for omoentropic flows, for isoentropic flows the Riemann invariant cannot be introduced and you need to integrate numerically the compatibility equations.