# 2D vs 3D Method of Characteristics for Rocket Nozzle Design

• BrandonBerchtold
In summary: NASA Marshall restricted so you might have to contact them to get a copy. It's a finite-rate chemistry code and uses a method of characteristics to generate the nozzle contour.
BrandonBerchtold
Hi all, I'm working on programming a simple 2D method of characteristics program to design the nozzle wall contour for a supersonic rocket nozzle. I'm wondering roughly what sort of difference I should expect from a 2D vs 3D method of characteristics program and where I could find a good description of the 3D method. Also, how do rocket companies go about designing their nozzle wall contours? Is the method of characteristics (2D or 3D) actually used by rocket companies or are there better methods for generating the nozzle contour?

Well, for one, the 2D version is only going to give you a proper contour for a 2D planar nozzle, i.e. one with a rectangular cross-section. If you want anything else, you will need to move to some 3D version. Thankfully, most of the time that means doing something axisymmetric since you want a circular cross section. An axisymmetric version will be less of a pain than full 3D.

There are a handful of codes out there that you can find in the public domain (though you'd likely have to manually transcribe them from old technical reports and papers into FORTRAN77 code. Otherwise, if you wanted to start from scratch, there are two references I can think of [1][2], both of which come in two volumes and have the 2D discussion in volume 1 and 3D discussion in volume 2.

While I don't work at a rocket company, it is my understanding that the method of characteristics is still the standard tool used to specify a supersonic nozzle contour. You would then need some kind of viscous correction. The method used in designing supersonic wind tunnels varies a little bit, but would look something like running MoC to get a contour, solving the boundary layer, and using the resulting displacement thickness to modify the contour to account for the boundary layers. You can iterate that process as well if desired. A rocket contour is similar, though the iteration is likely not as useful since wind tunnels need to be a lot more uniform in their exit flow than a rocket nozzle, which would be much more concerned with system weight.

[1] Shapiro, A. H. 1954. The dynamics and thermodynamics of compressible fluid flow. Vol. 2.
[2] Zucrow, M. J., Hoffman, J. D. 1977. Gas Dynamics. Vol. 2

BrandonBerchtold and berkeman
Perfect, that's just the answer I was looking for! I'll have to find me a copy of those references you recommended.

And that's very interesting, I was wondering how the boundary layer would be dealt with but simply modifying the contour based on the boundary layer thickness certainly makes sense.

Well, for one, the 2D version is only going to give you a proper contour for a 2D planar nozzle, i.e. one with a rectangular cross-section.
I'm pretty sure one can model a circular cross-section in 2D. 2D assumes azimuthal symmetry, where the central element(s) would be circular wedges and other elements annular sections.

About 35 years ago, I used a code from Marshall Space Flight Center, a fully coupled finite-rate chemistry simulation code, RAMP (Reacting And Multiphase Program), which uses Method of Characteristics, with another code, BLIMPJ (Boundary Layer Integral Matrix Program – Jet Version). We were modeling typical nozzles with circular cross-sections. 2D codes were used when there were significant memory (and bit) limitations in both storage and computational systems.

As far as I know, the are still used, and have been used with variable and constant specific heat ratio frozen flow simulations. Variations in turbulence models, temperature boundary conditions and thermodynamic properties of the plume have been investigated.

I believe the codes can be obtained from NASA Marshall subject to the usual restrictions.

Astronuc said:
I'm pretty sure one can model a circular cross-section in 2D. 2D assumes azimuthal symmetry, where the central element(s) would be circular wedges and other elements annular sections.

About 35 years ago, I used a code from Marshall Space Flight Center, a fully coupled finite-rate chemistry simulation code, RAMP (Reacting And Multiphase Program), which uses Method of Characteristics, with another code, BLIMPJ (Boundary Layer Integral Matrix Program – Jet Version). We were modeling typical nozzles with circular cross-sections. 2D codes were used when there were significant memory (and bit) limitations in both storage and computational systems.

Astronuc said:
I believe the codes can be obtained from NASA Marshall subject to the usual restrictions.

When discussing the method of characteristics as it pertains to nozzle design (and in aerodynamics more generally), 2D typically refers to "2D planar" and the cases with azimuthal symmetry are specifically termed "axisymmetric" to avoid conflation of the two. The distinction is important since the equations are different and they do not produce the same answer despite the fact that an axisymmetric set of equations are technically still two-dimensional.

RAMP is still used (not sure about BLIMP), but is not something that is available through the NASA software service (a lot of aero codes are if you are a US Citizen with a legitimate purpose). I did find the program manual for RAMP2, however, which indicates pretty clearly that it has a logical switch to toggle between 2D and axisymmetric cases within the code. So yes, RAMP can handle both, but it solves a slightly different set of equations depending on whether the flow is 2D planar or axisymmetric.

Astronuc said:
As far as I know, the are still used, and have been used with variable and constant specific heat ratio frozen flow simulations. Variations in turbulence models, temperature boundary conditions and thermodynamic properties of the plume have been investigated.

They are definitely still in use, though I am sure have undergone substantial changes over the years. How much longer that will be the case, who knows. They're probably written in FORTRAN77 (or earlier) and there are increasingly few people who know how to write or maintain FORTRAN code.

## 1. What is the difference between 2D and 3D Method of Characteristics for Rocket Nozzle Design?

The main difference between the 2D and 3D Method of Characteristics for Rocket Nozzle Design lies in the dimensionality of the design. The 2D method only considers two dimensions (length and width), while the 3D method takes into account the third dimension (height). This means that the 2D method simplifies the design process and may not accurately represent the actual flow behavior, while the 3D method provides a more realistic representation of the flow.

## 2. Which method is more commonly used in rocket nozzle design?

The 2D method is more commonly used in rocket nozzle design due to its simplicity and ease of implementation. It is also easier to visualize and analyze the results. However, the 3D method is becoming increasingly popular as computing power and simulation techniques improve, allowing for more accurate and realistic designs.

## 3. What are the advantages of using the 3D Method of Characteristics for Rocket Nozzle Design?

The 3D method provides a more accurate representation of the flow behavior, which can lead to more efficient designs. It also takes into account the effects of the third dimension, such as boundary layer growth and curvature, which can significantly impact the performance of the nozzle. Additionally, the 3D method allows for more complex nozzle geometries to be designed, which can further improve performance.

## 4. Are there any limitations to using the 2D Method of Characteristics for Rocket Nozzle Design?

Yes, there are some limitations to using the 2D method. As mentioned earlier, it simplifies the design process and may not accurately represent the actual flow behavior. This can result in less efficient designs and may require additional testing and adjustments to achieve optimal performance. Additionally, the 2D method may not be suitable for more complex nozzle geometries.

## 5. Can both methods be used together in rocket nozzle design?

Yes, both methods can be used together in rocket nozzle design. In fact, it is common for engineers to use a combination of both methods to take advantage of their strengths. The 2D method can be used for initial design and analysis, while the 3D method can be used for more detailed and accurate simulations. This approach can help optimize the design process and improve overall performance.

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