Space Exploration and Data Science

By Simon Mak and C.F. Jeff Wu

In his famous 1962 address at Rice University about U.S. space efforts to reach the moon, President John F. Kennedy declared that “the exploration of space will go ahead…and it is one of the great adventures of all time.” This adventure has become even more of a reality in recent years, as modern breakthroughs in mathematical modeling, scientific computing, and data science have inspired numerous pioneering achievements that were only dreams in the past.

A crucial part of space exploration is the design and development of high-performance rocket propulsion systems, and a central component of such systems is the rocket injector. This device enables the intermixing of fuel and oxidizer prior to combustion; Figure 1 displays a schematic of a simplex swirl injector [1, 2, 6, 10]. Liquid oxygen is injected into a gaseous fuel environment via the inlet slots on the left, then swirled along the injector wall and propelled out for combustion. The design space consists of \(p=5\) parameters: injector length \(L\), injector radius \(R_n\), inlet slot width \(\delta\), tangential inlet angle \(\theta\), and the distance between the inlet and the headend \(\triangle L\) (one can easily extend the methods that we propose for more complex design settings with additional parameters). The goal is to identify parameter designs that ensure good mixing of fuel and oxidizer, which translates to a fuel-efficient and robust combustion performance.

In order to explore potential injector designs, engineers must conduct experiments over the desired parameter space. Traditional physical experiments are prohibitively expensive due to high prototyping costs and the harsh requirements of operating conditions [4]. Because these experiments rely on optical diagnostics for measurements, they offer minimal insight into the underlying physiochemical mechanisms [9]. Due to recent advances in scientific modeling, numerical simulations are becoming a reliable alternative to physical tests. These so-called computer experiments can provide more salient features of the flow and combustion dynamics within the injector [7]. They also allow for significant savings in prototyping and experimental costs.

To reliably capture the rich physics within the turbulent flow, a sophisticated multiphysics computational fluid dynamics (CFD) model is necessary for simulation purposes. We begin with the well-known Navier-Stokes equations, which express the conservation of mass and momentum for Newtonian fluids. Then we model thermodynamic properties via the Soave-Redlich-Kwong equation of state, which correlates fluid pressure, temperature, and density under high-pressure conditions. The large eddy simulation (LES) framework achieves turbulence closure. Figure 2 depicts instantaneous snapshots of the simulated temperature and density flows of an injector design.

However, a key limitation of CFD-based design exploration is that these flow simulations are computationally expensive. For a fine grid of 100,000 mesh points, a high-fidelity flow simulation that uses LES can take roughly 30,000 central processing unit (CPU) hours to obtain statistically meaningful data. This restriction greatly limits the number of potential designs that are sampled on the parameter space, particularly given the number of geometric attributes and operating conditions for survey. We utilized a surrogate modeling approach to address this limitation. Surrogate modeling consists of two steps. One first performs simulations at a carefully selected set of design points on the parameter space, then fits a predictive model using the simulated flows as training data. The fitted model serves as a surrogate for the expensive LES code, thus enabling efficient exploration of the parameter space.

We employed the proper orthogonal decomposition (POD) to build this flow surrogate model [1, 2, 6, 10]. Researchers frequently employ the POD in experimental physics to decompose a turbulent flow into its component coherent structures. When \(f_\theta(s,t)\) is the simulated flow at design setting \(\theta\)—where \(s\) and \(t\) are spatial and temporal variables—the POD yields the following decomposition:

\[f_\theta(s,t) \approx \bar{f}_\theta(s)+\Sigma^K_{k=1}\beta_{k,\theta}(t)\phi_{k,\theta}(s).\]

Here, \(\bar{f}_\theta(s)\) is the time-averaged flow at design setting \(\theta\), \(\mathcal{S}(\theta)=\{\phi_{k,\theta}(s)\}^K_{k=1}\) is the set of orthonormal spatial modes, and \(\mathcal{T}(\theta)=\{\beta_{k,\theta}(t)\}^K_{k=1}\) is the set of time-varying coefficients. We can extract these spatiotemporal POD features via a singular value decomposition from the simulated flow snapshots.

To predict the flow \(f_{\theta_\rm{new}}(s,t)\) at a new design setting \(\theta_{\rm{new}}\), we first build two surrogate models: one for predicting the spatial modes \(\mathcal{S}(\theta_{\rm{new}})\) and one for predicting the time-varying coefficients \(\mathcal{T}(\theta_{\rm{new}})\). We construct both models using Gaussian processes—a flexible Bayesian nonparametric predictive model—and train them with the extracted spatiotemporal POD features from simulated flows. We can utilize decision trees [10] and kernel smoothing methods [2] to further learn the changing physics over the parameter space, such as the boundary between jet and swirl injectors. Once we predict the spatial modes and time-varying coefficients, we can then reconstruct the corresponding flow prediction at the new design setting \(\theta_{\rm{new}}\) via the aforementioned decomposition.

Next, we examine the performance of this flow surrogate model on four test injector designs that were taken over a broad parameter space that contains the RD-0110 [8] and RD-170 [3] engines. We trained our model with flow simulations at \(n=30\) design settings from a sliced Latin hypercube design. Figure 3 presents the simulated flow snapshots from LES and predicted flow snapshots at different times. The predictions appear to successfully capture the large-scale features of the instantaneous flow, including the spray angle and the liquid film along the injector wall. The key advantage of this surrogate model is the savings in computation time. After simulating the initial \(n=30\) training runs, we can train our model using 150 CPU hours. With the trained model, we can generate the flow prediction at a new design setting in approximately 30 CPU hours. This outcome provides significant (1,000-fold) computational reduction when compared to a full CFD simulation, which requires 30,000 CPU hours per setting; it thereby allows engineers to more efficiently explore the desired parameter space.

These results, while promising, likely only scratch the surface of an exciting and emerging interdisciplinary field. In order to tackle the unavoidable challenges of this ambitious space travel adventure, practical solutions must reflect sophisticated developments and integration of state-of-the-art methods in scientific modeling, applied mathematics, and data science. Such interdisciplinary efforts collectively serve as a catalyst for furthering scientific progress and in turn spur novel methodological development.

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References
[1] Chang, Y.-H., Wang, X., Zhang, L., Li, Y., Mak, S., Wu, C.F.J., & Yang, Y. (2021). Common kernel-smoothed proper orthogonal decomposition (CKSPOD): An efficient reduced-order model for emulation of spatiotemporally evolving flow dynamics. Preprint, arXiv:2101.08893.
[2] Chang, Y.-H., Zhang, L., Yeh, S.-T., Wang, X.J., Mak, S., Sung, C.-L., …, Yang, V. (2019). Kernel-smoothed proper orthogonal decomposition (KSPOD)-based emulation for prediction of spatiotemporally evolving flow dynamics. AIAA J., 57(12), 5269-5280.
 [3] Dranovsky, M.L. (2007). Combustion instabilities in liquid rocket engines: Testing and development practices in Russia. In Progress in astronautics and aeronautics (Vol. 221). Reston, VA: American Institute of Aeronautics & Astronautics.
[4] Lieuwen, T.C., & Yang, V. (2013). Gas turbine emissions. Cambridge, UK: Cambridge University Press.
[5] Lumley, J.L. (1967). The structure of inhomogeneous turbulent flows. In A.M. Iaglom & V.I. Tatarski (Eds.), Atmospheric turbulence and radio wave propagation (pp. 221-227). Moscow, Russia: Nauka Publishing House.
[6] Mak, S., Sung, C.-L., Wang, X.J., Yeh, S.-T., Chang, Y.-H., Joseph, V.R., …, Wu, C.F.J. (2018). An efficient surrogate model for emulation and physics extraction of large eddy simulations. J. Am. Stat. Assoc., 113(524), 1443-1456.
[7] Poinsot, T., & Veynante, D. (2005). Theoretical and numerical combustion (2nd ed.). Philadelphia, PA: R.T. Edwards, Inc.
[8] Wang, X., Wang, Y., & Yang, V. (2017). Geometric effects on liquid oxygen/kerosene bi-swirl injector flow dynamics at supercritical conditions. AIAA J., 55(10), 3467-3475.
[9] Yang, V., Habiballah, M., Hulka, J., & Popp, M. (2004). Liquid rocket thrust chambers: Aspects of modeling, analysis, and design. In Progress in astronautics and aeronautics (Vol. 200). Reston, VA: American Institute of Aeronautics & Astronautics.
[10] Yeh, S.-T., Wang, X.J., Sung, C.-L., Mak, S., Chang, Y.-H., Wu, C.F.J., & Yang, V. (2018). Data-driven analysis and common proper orthogonal decomposition (CPOD)-based spatio-temporal emulator for design exploration. AIAA J., 56(6), 2429-2442.

Simon Mak is an assistant professor in the Department of Statistical Science at Duke University. His research—which lies at the intersection of Bayesian methodology, machine learning, and scientific computing—is motivated by ongoing projects in engineering and nuclear physics. C.F. Jeff Wu is the Coca Cola Chair in Engineering Statistics at the Georgia Institute of Technology. His current research interests are in developing statistical methods that are motivated by and applied to problems in engineering and physical sciences.