DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators
Introduction
Simulating the high-speed flow field of a chemically reacting fluid is interesting, challenging and has important applications including hypersonic cruise flights and planetary re-entry [1]. In order to predict such a flow field, a multi-physics and multi-scale approach is essential. The fluid dynamics part of this problem may already feature a large number of effects as well as length and time scales when it features shocks, boundary layers, transition to turbulence to name but a few [49], [4], [13], [7]. In addition to fluid dynamics, also physical chemistry needs to be taken into account, as chemical reactions are likely to occur in the flow, and these need to be modeled accurately [37], [2], [44]. It is particularly challenging if reactions and flow effects take place on similar spatio-temporal scales because a simplified model for this type of flow field cannot be easily derived. Instead, the full set of equations describing the physical as well as the chemical model must be solved.
One particularly relevant canonical problem is the high-speed flow downstream of a normal shock [28]. In this case, the temperature of the fluid rises rapidly across the shock, which in turn triggers chemical dissociation reactions. As a result, the flow field changes composition, which influences the energy balance as dissociation reactions are endothermic. A change of composition directly influences several other aspects because it leads to a modification of viscosity as well as heat conduction.
The level of fidelity required to model flows with high-temperature gas effects depends on the typical flow speed and temperature [1], [37], [39]. The rate of chemical reactions is largely influenced by temperature (and to a lesser extent by pressure), and convective mass transport is driven by flow speed. Comparing the typical time scales of these two effects yields three different regimes. In the first, chemical reaction rates are much smaller than the rate of convective transport, and the fluid composition in this regime is considered frozen. If chemical reaction rates are much larger than the rate of convective mass transport, the flow is in chemical equilibrium; reaction rates are therefore assumed infinite and the gas composition depends on local properties such as temperature and pressure (or density). The most interesting regime, which is considered herein, is referred to as finite-rate chemistry, or non-equilibrium chemistry; it lies in between the other two regimes, when the rates of reactions and transport are commensurate.
Gas composition strongly affects the relation between temperature and internal energy. For a calorically perfect gas, internal energy is proportional to temperature. A gas composed of a single atomic species typically behaves calorically perfect, for which the specific heat is constant. A gas composed of a molecular species will experience the excitation of vibrational and electronic modes of the molecules. As a result, the internal energy becomes a non-linear function of temperature and the specific heat will also vary with temperature. Such a gas is denoted as thermally perfect. Vibrational and electronic excitation may not happen infinitely fast, in which case the process of thermal relaxation may need to be taken into account. However, here we will only consider a gas in so-called local thermal equilibrium, i.e., it will be assumed that the vibrational excitation only depends on local properties such as the temperature and not on its (time) history.
In many practical applications, the fluid is a mixture of atomic and diatomic species, such as in air. While air can certainly be modeled as a calorically perfect gas at low temperatures where vibrational and electronic modes are not excited, at high temperatures a thermally perfect gas model would be more appropriate. At even higher temperatures, the threshold for the onset of chemical reactions may be reached and the corresponding changes of gas composition may need to be included in the modeling approach. Such a high temperature may occur as a result of a shock wave, which creates an almost instantaneous increase of temperature [21], [19]. The time scale for a flow passing through a shock is extremely short, owing to the very small thickness of a shock, which is on the order of the mean free path lengths of the species involved. Chemical reactions are much slower than this fast time scale, and hence the composition of the gas mixture does not change across the shock: the gas mixture is frozen. However, downstream of the shock, chemical reactions may set in as a result of the higher temperature, and as the flow speed is reduced significantly, a region of flow in the finite-rate reaction regime is expected, before the flow may reach an equilibrium state far downstream of the shock. As a result of finite-rate reactions, the gas mixture will increasingly change its composition as molecules begin to dissociate, rendering the region immediately downstream of the shock particularly interesting.
Compared to a calorically perfect gas, a gas that undergoes changes in composition is appreciably more complex to model because transport equations must be solved for each species density, with a source term for the creation and destruction of species by chemical reactions. These additional equations complicate the numerical treatment significantly. In particular, the source term can cause numerical stiffness and hence become difficult to integrate. Several numerical methods for finite-rate chemistry exist, mostly for time-dependent flows involving combustion [15], [34], [10], [33], [35], but they often rely on a low-Mach-number formulation and are therefore not applicable to hypersonics. At high Mach number, evidence abounds regarding the sensitivity of the flow to small distortions [40], [16], and hence accurately capturing non-equilibrium chemistry becomes extremely important. A growing number of methods now exist that account for non-equilibrium chemistry while solving the compressible Navier-Stokes or Euler equations at hypersonic speeds [52], [18], [30], [12], [41], [8], [50], [51]. However, the combination of high Mach numbers and high temperatures not only renders these simulations challenging, but also taxes computational resources heavily.
In realistic hypersonic applications, flight data may be comprised of limited measurements, for example of temperature or velocity and perhaps even in special cases of the composition of the species. To integrate such data with the computational approach and in order to simulate the aforementioned multiscale & multiphysics problems efficiently, we abandon the classical numerical methods and explore in the present work a new approach, namely, a deep neural network (DNN) based approximation of all nonlinear operators.
The machine learning community has made tremendous strides in the past 15 years by capitalizing on the neural network (NN) universal function approximation [9], and building a plethora of innovative networks with good generalization properties for diverse applications. However, it has ignored an even more powerful theoretical result by Chen & Chen [5], [6] that states that neural networks can, in fact, approximate functionals and even nonlinear operators with arbitrarily good accuracy. This is an important result with significant implications, especially for modeling and simulation of physical systems, requiring accurate regression and not only approximate classification tasks as in commercial applications. Preliminary results in [11], [23] have provided a glimpse of the potential breakthroughs in modeling complex engineering problems by encoding different explicit and implicit operators using DNNs. For example, in [11] Ferrandis et al. represented a functional predicting the dynamic motions of a destroyer battleship in extreme sea states, making predictions at a fraction of a second in contrast to one week per simulation using OpenFoam CFD solver. Similarly, in [23], Lu et al. developed the Deep Operator Network (DeepONet) to approximate integrals, ODEs, PDEs, and even fractional Laplacians by designing a new trunk-branch NN that approximates linear and nonlinear operators, and generalizes well to unseen functions.
Traditional methods, especially high-order discretizations such as WENO [22] and spectral elements [17], can produce very accurate solutions of multiphysics and multiscale (M&M) problems but they do not scale well in high dimensions and large domains. Moreover, they cannot be easily combined with data [46], [47], [31], [3] and are prohibitively expensive for inverse problems. Real-world M&M problems are typically ill-posed with missing initial or boundary conditions and often only partially known physics, e.g., reactive transport as in the present work. Physics-Informed Neural Networks (PINNs) can tackle such problems given some extra (small) data anywhere in the domain, see [42], [43], [27], [24]. PINNs are easy to implement for multiphysics problems and particularly effective for inverse problems [36] but not as efficient or accurate for forward multiscale problems. Here, we propose DeepONets to approximate functionals and nonlinear operators as building blocks of a more general M&M framework that can be used to approximate different nonlinear operators for modeling M&M problems. Unlike PINNs, we can train DeepONets offline and make predictions for new input functions online very fast. We refer to this integrated framework that will use both data and DeepONets as DeepM&Mnet, and, in principle, it can be used for any complex M&M problem in physics and engineering. Here we consider hypersonic flow downstream of a normal shock, which involves the interaction of seven field variables (see Fig. 1). This formidable M&M challenge is an excellent testbed to develop the DeepM&Mnet framework and to demonstrate its effectiveness.
The aim of this work is to develop a deep-learning framework using neural-network approximation for solving the coupled flow and finite-rate chemistry in hypersonics flows. We first build connections between the flow and the chemical species, namely, we build functionals approximated by neural networks between the flow and chemical species (i.e., taking the flow as the functional of chemical species or taking chemical species as the functional of the flow) with DeepOnets, which will serve as building blocks for the DeepM&Mnet. We then build parallel or series DeepM&Mnets, which take the space variable x as input and field variables as outputs, and train these networks by using the predictions (which are required at each step of the training) of the pre-trained DeepOnets between the flow and chemical species. In a DeepM&Mnet, we first train several DeepONets independently as the subcomponents, and then train one extra network, which shares a similar idea of transfer learning [45]. In particular, in the present work, we claim the following contributions:
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We start with developing DeepONets for the M&M model, namely, the non-equilibrium chemistry that takes place behind a normal shock at Mach numbers between 8 and 10. We infer the interactions of the flow and five chemical species whose densities span 8 orders of magnitude downstream of the shock. Collectively, these dynamics establish the gas composition and flow velocity, which are governed by the nonlinear Navier-Stokes equations and whose operators will be learned by our DeepONets. Performing inference on a trained DeepONet allows evaluation of the solution in 100000x less time compared to a traditional CFD solver.
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Besides the prediction for the case when the Mach number is in the range , we also test the case when the input is out of the input space, i.e., the Mach number is out of the range (extrapolation). There are relatively large deviations between the reference data and the predictions obtained by directly using the DeepONets. However, we significantly improve the predictions and obtained good results by combining a few data, which may be available from sensor data, and the pre-trained DeepONets by developing a supervised NN, which can be efficiently trained.
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As a preliminary step in developing the multi-physics integrated framework, we employ these pre-trained DeepONets as building blocks to form different types of DeepM&Mnets. We first develop a parallel DeepM&Mnet architecture which, similar to the aforementioned extrapolation algorithm, requires sensor data for all the variables. However, in practice, for data assimilation we may not have access to information regarding the species densities, and may only have sparse data for the flow. Therefore, we develop a series DeepM&Mnet architecture that assimilates only a few data for the flow and predicts the entire state. Moreover, we examine the influence of the global mass conservation constraint and demonstrate that not only does it stabilize the training process but it also improves prediction accuracy.
The rest of the paper is organized as follows: In the next section, we present the M&M fluid-mechanical model and demonstrate how to generate the training data using a finite difference approach. We then develop in section 3 the DeepONets, which will serve as building blocks for the DeepM&Mnets discussed in section 4. We conclude with a summary in section 5. In the Appendix we present an alternative series type DeepM&Mnet.
Section snippets
Problem setup and data generation
In this section, we present in detail the governing equations that model the flow and describe how to obtain the data for training and testing. The mathematical formulation for fluid motion is given in § 2.1, followed by a description of the data generation process including details of the test case in section § 2.2.
Developing DeepONets as building blocks
The main idea of the proposed framework is to map an input in the form of a function to an output in the form of another function. This is accomplished by using the DeepOnet that expresses operator regression, which is different from a standard neural network that regresses a function. In this section, we present the DeepONet architecture, which will serve as building block for the DeepM&Mnet. We develop DeepONets for different fields to express the coupled dynamics between the flow and the
DeepM&Mnet framework: architectures and results
In this section, we propose the DeepM&Mnet framework for hypersonics multi-physics and multiscale problems, by coupling the pre-trained DeepONets and developed in subsection 3.2. Unlike the building blocks DeepONets that require input functions and make predictions, we now relax this requirement. We only assume that we have some sensor data for the inputs. For all the tests performed in this section, we randomly select a value of the Mach number in the interval .
Conclusion
The simulation of hypersonic flow is a challenging multi-scale & multi-physics (M&M) problem, due to the combination of high Mach, the interaction with a shock leads to excessively high temperatures that can cause dissociation of the gas. When the reaction rates are commensurate with the rates associated with the flow itself, the dynamics of the dissociation chemistry and the flow are coupled and must be solved together. These coupled physics lead to changes in the chemical composition of the
CRediT authorship contribution statement
Zhiping Mao: Conceptualization, Methodology, Investigation, Coding, Writing - original draft, Writing - review & editing, Visualization.
Lu Lu: Conceptualization, Methodology, Investigation, Writing - original draft, Writing - review & editing.
Olaf Marxen: Providing DNS data, Writing - original draft, Writing - review & editing.
Tamer A. Zaki: Conceptualization, Methodology, Investigation, Writing - original draft, Writing - review & editing, Supervision, Project administration, Funding
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors acknowledge support from DARPA/CompMods HR00112090062.
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This work started while Z. Mao was a postdoc at Brown University.