Computational Optimization of Rocket Engine Nozzles

By Konstantinos Kontogiannis
Supervised by Dr. Graham T. Roberts
Individual Project – MSc Aerodynamics & Computations
Faculty of Engineering and the Environment
University of Southampton

The primary aim of this project was to investigate the capability of using modern commercial computational fluid dynamics (CFD) packages (Pointwise and ANSYS Fluent®) as tools for analysis and design of rocket engine nozzles. It should be noted that since the project is not targeted at a specific nozzle application, its main purpose is to demonstrate and give more insight and guidance on rocket engine nozzle design utilizing modern CFD and design search and optimization techniques.

The first step was establishing a method to analyze the performance of a contoured axisymmetric nozzle. This included creating the appropriate geometry, meshing the flowpath, setting up the CFD solver, performing a mesh convergence study, validating the results and finally trying to minimize the computational cost. Afterward, a number of investigations on nozzle design methodologies were conducted using conical nozzles and thrust optimized Rao parabolic nozzle contours.

A number of codes in C++ were written in order to be able to generate such contours given the required design parameters. These codes were then utilized in automating the design evaluation process so it could be coupled with the design optimization algorithm. This included automating the geometry generation according to design parameter values provided by the optimization algorithm, then importing that geometry to the mesh generation software and creating the corresponding computational grid, running the CFD solver and finally post-processing the results to be fed back into the optimization software. This automation is necessary as it would be much more time consuming and error-prone for the user to manually do this for every design, especially the mesh generation step.

Two cases were studied, the first one being a multi-objective optimization of a conical nozzle with the design variables being the diverging cone half angle and the area ratio with the objectives of maximizing the performance while trying to minimize the size of the nozzle for operation in vacuum conditions. The second case was based on displacing a parabolic Rao nozzle contour [Reference 1] as shown in Figure 1 with a Bezier polynomial using two control points in order to improve its performance.

Detailed analysis methods, although more time consuming and computationally expensive, are possible with today's advanced computer systems. They are also more robust compared to methods used in the past, given that the design objectives or constraints can be changed and more than one physical mechanism can be taken into account with a coupled, multi-disciplinary analysis.

structured grid for a 2D nozzle
Figure 1: This structured 2D grid for a Rao parabolic nozzle was generated by Pointwise and imported into ANSYS Fluent.

Although the mesh is fairly simple, the important point was to automate the process so that an appropriate mesh is generated for any geometry specified by the design optimization algorithm. Mesh generation can be the part of the process where most difficulties arise. However, this task proved fairly easy because of the relatively simple nature of the grid, the straightforward way to generate the grid in Pointwise and because of Pointwise's ability to record a Glyph script that was ready to be used in the design optimization loop after some minor alterations.

After having scripted/automated the design evaluation method and having chosen the parameterization of the design, an initial sample was generated using an optimized Latin Hypercube method and all the designs were subsequently evaluated. QstatLab software was used with a kriging method to create a surrogate model. The response of the nozzle equivalent exit velocity to two of the design variables is shown in Figure 2, while Figure 3 shows the initial and final optimized nozzle shapes.

This initial successful study has shown the possibilities of coupling Pointwise and ANSYS Fluent with design optimization software to generate rocket nozzle designs meeting multiple objectives. In the future, this can be expanded to a more complete nozzle design method by implementing more complex CFD techniques or including a FEA analysis utilizing the CFD data (pressure distribution, heat fluxes, etc.) as input.

plot of exit gas velocity variation with design variables
Figure 2: Contours of the kriging surrogate model of the nozzle equivalent exit gas velocity show how it varies with the two design variables. Exact evaluations are shown in white, and the initial design is at (0,0).
copmarison of initial and optimized nozzle contours
Figure 3: The initial (pink) and final (purple) nozzle geometries are overlaid in this image.

[Reference 1] Rao G. V. R. (1960), “Approximation of Optimum Thrust Nozzle Contour”, ARS Journal, Vol. 30, No. 6, p. 561.

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