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Reconstructing cellular surface of gaseous detonation based on artificial neural network and proper orthogonal decomposition
Combustion and Flame 212 (2020) 156–164
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Combustion and Flame journal homepage: www.elsevier.com/locate/combustflame
Reconstructing cellular surface of gaseous detonation based on artificial neural network and proper orthogonal decomposition Yining Zhang a, Lin Zhou a,b, Hao Meng a, Honghui Teng b,∗ a b
State Key Laboratory of Laser Propulsion & Application, Beijing Power Machinery Research Institute, Beijing 100074, China School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 22 August 2019 Revised 22 October 2019 Accepted 23 October 2019
Keywords: Cellular detonation Oblique detonation Proper orthogonal decomposition Artificial neural network
a b s t r a c t Gaseous detonation has complicated cellular surface, whose comprehensive investigation is critical not only to the detonation physics but also the detonation engine development. Because measuring the highresolution dynamic surface is beyond the present experimental technical skills, we propose a reconstruction method of detonation wave surface based on post-surface flow field. This method combines two technologies, the proper orthogonal decomposition (POD) in fluid research and the artificial neural network (ANN) in machine learning research. POD is employed to extract the main features of flow fields, and the pre-trained ANN builds up the connection between the reduced coefficients of full flow fields and post-surface flow fields. The reconstruction is tested through the numerical results from one-step irreversible heat release model, displaying a good performance in both cellular normal detonations and unstable oblique detonations. The method may provide a universal frame for the detonation research, and has the potential to be employed in other numerical and experimental results. © 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Gaseous detonation is an extreme combustion phenomenon and has been studied widely for its extensively existing in catastrophic explosion and astrophysics [1–3]. Recently, detonation research on its astronautic propulsion attracts more and more attention to develop high-efficiency engines [4,5]. The new trend needs underlying understanding on the wave dynamics, which provides the foundation for the engine design and adjusting the waves inside. As shock-induced combustion, the detonation dynamics is dominated by the surface composed of shock-flame complex. Different from classic ZND (Zeldovich-von Neumann-Döring) model prediction, realistic wave surfaces are cellular due to reciprocating transverse waves [6]. Hence, cellular structures become one of the most important direction in the detonation physics research, such as [7–13]. Recording tracks of triple points through smoked foils has been the widely used technique to obtain the evolution history of the detonation wave structure in past decades [14–18]. However, in the detonation engine research, the wave surface is usually treated as a smooth one following the ZND model [19,20]. Recent numerical study [21,22] demonstrated that the non-premixed inflow results in a very complicated wave system in rotating detonation en-
∗
Corresponding author. E-mail address: [email protected] (H. Teng).
gines (RDE), but very limited experimental results, such as [23,24], are available due to the difficulty of measurement technology. This is because the highly unstable surface, deriving from cellular shock-flame complex, brings much uncertainty on the quantitative measurement. To overwhelming these difficulties, an alternative approach is measuring the post-surface flow field directly, and reconstructing the surface based on the measured information and machine learning technique. Since the post-surface region has relatively large time and length scales, this indirect method has much lower technical challenge than measuring wave surface directly. A reconstruction method of detonation wave surface based on the post-surface flow field is proposed in this study. The key idea is to establish a connection between full flow field of gaseous detonation and corresponding post-surface flow field using an artificial neural network (ANN). As an extensively used data-driven technique, ANN has shown its powerful ability to learning the underlying mechanism from training by large data set [25–28]. Considering the experimental results are limited and as the first step, the numerical results from one-step irreversible heat release model are used here to check the feasibility and problems of the proposed method. To be used in ANN, the features of shock-flame complex should be extracted first, which has been performed through proper orthogonal decomposition (POD). POD method is a very effective feature extraction and dimension reduction method, which has been widely used in turbulence and combustion fields [29–32]. ANN and POD methods have been used collectively in
https://doi.org/10.1016/j.combustflame.2019.10.031 0010-2180/© 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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flow filed reconstruction to relieve the curse of the dimensionality in aerodynamic [33]. However, this proposed method has the essential difference because of the reconstruction from post-surface flow field, rather not simplified problem. The method is introduced in Section 2 in detail, and two serial cases are tested in Section 3, demonstrating a good ability in both cellular normal detonations and unstable oblique detonations. The conclusion and discussion on future work are given in Section 4. 2. Reconstruction method 2.1. Introduction on proper orthogonal decomposition The reconstruction method actually builds up a mapping from post-surface flow field to the full flow field, and the mapping of this study is achieved through an ANN. However, describing the flow field requires large amounts of data, e.g., the parameters in each mesh, so the flow field cannot be used directly as the input and output of ANN. Fortunately, there are still some fixed basic modes (features) of detonation flow fields despite of the transient flow field varying at different times, so different combinations of these modes can constitute the transient flow field. The POD is a method of decomposing the transient flow field sequence into linear combinations of a set of fixed basic modes along with timevarying coefficients [34]. Thereafter, only the coefficients of these modes at certain time need to be known to reconstruct the corresponding flow field. The optimality of the POD reduces the amount of information required to represent statistically dependent data to a minimum [35]. Moreover, the POD also captures the essential dynamics of the flow and provides insight into the nature of the flow and its instabilities [36]. In conclusion, the POD here is used to compress flow field efficiently with representing flow field with fixed modes and corresponding coefficients, so favoring the ANN training. For simulated flow fields, the method of snapshots [37,38] is used to implement the POD. According to the simulation grid, the transient flow field can be arranged into a n-dimensional vector si in rows or columns order. Then the si at m different instants can be arranged in a sequence to form a temporal snapshots matrix S = {s1 , s2 , . . ., sm }. Each snapshot si contains a quantity of scalar flow magnitudes which in the case of experimental results usually represents the velocity information (PIV (particle image velocimetry) technology). In the case of numerical simulations these snapshots may represent other useful information such as pressure, temperature, species, etc. In this paper, pressure field extracted from the simulation results is used as snapshot si to represent the wave structure of detonation flow field. Then, we compute the reduced singular value decomposition (SVD) of snapshots matrix S:
S = U V
T
(1)
Here, U is an n × r matrix whose columns consist of the POD modes vectors uj . is an r × r diagonal matrix whose diagonal components are the singular values σ j of S. They describe the energy contained in each mode which indicating the relevance of each mode in the total flow field [39]. V is an m × r matrix and each component of the row vector ci of VT describes the contribution of each mode uj to the snapshot flow field si . Thus, si can be represented as a linear combination of fixed basic modes uj according to the coefficient vector ci with k = r:
si =
k
u j ci j
(2)
j=1
To further relieve the ANN designing and training, si can be approximated following Eq. (2) but with a reduced k < r number of
Fig. 1. Basic architecture of multiply feedforward ANN.
modes. The number of retained modes k used to reconstruct the full flow field depends on the reconstruction accuracy. 2.2. Introduction on artificial neural network After the flow fields are decomposed into the fixed modes and corresponding coefficients by POD, the next step is using ANN to establish the mapping from the reduced modes coefficients of post-surface sub-field to those of the full flow field. The basic architecture of typical multiply feedforward ANN is shown in Fig. 1. The network includes one input layer, several hidden layers, and one output layer. In each layer, there are several neurons which connect with neurons of other layers. Unfortunately, there are no universal and effective rules to determine the detailed number of hidden layers and neurons in ANN research fields. In this work, we use two hidden layers, whose neurons are varied depending on the reconstruction problems. The Nguyen-Widrow layer initialization function is used as the ANN initial method [40], and weight and bias values of neurons are updated according to the scaled conjugate gradient algorithm [41]. The scaled conjugate gradient algorithm is based on conjugate directions, but a step size scaling mechanism is used to avoid a time-consuming line-search per learning iteration, making the algorithm achieve better superlinear convergence rate. The performance function used to evaluate the ANN is mean-squared error along with the regularization term, which is used to relieve overfitting. Moreover, validation is used to stop training early if the network performance on the validation set fails to improve or remains the same for certain pre-set epochs. 2.3. Algorithm of flow field reconstruction By combining the POD and ANN, the full field can be reconstructed from the post-surface sub-field. A flowchart illustrating the present algorithm of flow field reconstruction is shown in Fig. 2. The full flow fields and corresponding sub-fields snapshots set are extracted first from the transient flow fields based on the simulation results to carry out the training of ANN. It is worth noting that, to ensure the reconstruction accuracy, the POD modes of sub-fields and full flow fields from the training and validation set should be basically consistent to those from the test set. That is, it’s necessary that the detonation wave can complete several cycles of change process during the time period of extracting transient flow field. Thus, the training set can contain typical dynamic features of transient wave structure as much as possible. Then, decomposing both the full fields and sub-fields of training and validation set into basic modes and corresponding coefficients set through POD according to Eq. (1). Thereafter, determining the number of full flow field modes need to be retained to satisfy the required accuracy of reconstruction, which corresponding to
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Fig. 2. Flowchart of the wave surface reconstruction algorithm based on ANN and POD.
the number of neurons No in the output layer of ANN. As for the input layer neurons number Ni , generally speaking, the more subfield information provided to ANN input, the more accurate the reconstructed result will be. On the other side, too much input will lead to the training of ANN difficult. Since the sub-fields usually have relatively slow time- and space-varying flow field structure, the corresponding retained POD modes, i.e., the input layer neurons number Ni , are determined to be less than those of the full fields, i.e., the output layer neurons number No . In this work, we conduct several trial calculations to determine the detailed number of neurons in the input layer and two hidden layers of ANN, which are Ni , Nl1 and Nl2 , respectively. In practice, we gradually increase the number of neurons and do various trainings, the network with the smallest error is chosen to be the final model. Furthermore, the number of neurons in the hidden layers are chosen to be close to the input and output layer neurons number, which can be helpful to construct a simple ANN. Finally, we train the ANN with both training and validation set to establish the mapping relationship between the first Ni coefficients corresponding to the post-surface sub-fields modes and the first N° coefficients corresponding to full flow fields modes. Once the time-consuming training finished, reconstructing the full flow field using the trained ANN requires only simple and fast algebraic computation. For any post-surface sub-field A in the test set, the corresponding sub-field modes coefficients can be easily calculated by using the known sub-fields POD modes. Then, the well-trained ANN is used to compute the coefficients corresponding to the full flow field modes of A. Finally, the reconstructed full flow field A can be obtained by the linear superposition of the known full flow field POD modes according to Eq. (2). In order to evaluate the reconstruction performance quantitatively, the average relative error between the reconstructed full field and the real full field is defined as follows:
Error =
M N 1 P˜m,n − Pm,n Pm,n × 100% MN
(3)
m=1 n=1
Here Pm,n and P˜m,n denote the real and reconstructed pressure in the grid point (m, n), respectively. M and N denote the number of reconstructed flow field points in the x- and y- direction.
3. Results and discussion 3.1. Cellular normal detonations The proposed method is first used to reconstruct the cellular surface of normal detonation propagating in a rectangular tube. First of all, cellular detonation flow fields at different instants, like those shown in Fig. 3, were simulated using the numerical methods introduced in Appendixes. Then, a data set consisting of 300 transient detonation flow fields is extracted after the general detonation structure reaches a quasi-steady state. The data set is divided randomly as the training set and validation set, with the ratio 85% and 15%, and additional 50 different transient detonation flow fields are further generated from other transient flow fields as test set. The extracted region has the height of 12, with the leading shock kept near 10. In the training and test processes, the sub-field has the same width with full field, while the height is chosen to be 2/3 of the full field, that is 8 from the bottom. Application of the POD to the 300 transient cellular detonation full field snapshots yields a set of modes with each mode indicating some inner features of the flow field. The first six POD modes are shown in Fig. 4. The first mode is the main mode which stands for the averaged macroscope leading shock wave structure of the cellular detonation wave surface. Mode 2 to 6 show the different shock-flame complex composed of leading shock and transverse waves. Different combinations of all modes can exhibit the complex dynamic process of the cellular detonation wave. To determine how many modes are needed to reconstruction full flow field, we project typical flow fields onto the different retained modes, and use Eq. (2) to recover the approximated solutions, shown in Fig. 5. As expected, the approximation to the CFD solution improves with increasing retained modes of reconstruction, and 30 modes retained is able to generate a good approximation. Thus, the ANN output layer neurons number is set to be 30, and other ANN parameters used in the cellular normal detonations case are shown in Table 1. Using the well-trained ANN, we reconstruct the full field from the post-surface sub-field, and a test case including totally 50 flow fields chosen randomly from simulation results is examined to evaluate the method’s performance. The average relative errors calculated by Eq. (3) are shown in Fig. 6. Obviously, more than half of the reconstruction errors are smaller than 10%, and the average reconstruction error of this test set is about 9.39%, which is accept-
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Fig. 3. Cellular normal detonation wave surface at different moments.
Fig. 4. The first six POD modes of cellular normal detonation wave.
Fig. 5. POD approximations with different reconstruction precision.
Table 1 ANN parameters for the cellular normal detonations case. Layers
Neurons number
Input layer Hidden layer 1 Hidden layer 2 Output layer
24 25 35 30
able considering several parameters can be optimized in the future work. To analyze the reconstruction results, some typical flow fields are shown in Fig. 7. The first one in Fig. 7(a) corresponds to the 11th transient flow field in Fig. 6, with the relative reconstruction error 9.45%, close to the average error of the test set. It can be seen that the triple points and transverse waves are reproduced clearly, except that the wave surface has a little bit blurred without affecting the distinguished cellular structure. The reconstructed flow field with the largest reconstruction error 17.42% in the test set is shown in Fig. 7(b), corresponding to the 4th transient flow field in Fig. 6. In contrast, Fig. 7(c) corresponds to the 23rd transient flow field in Fig. 6, with the relative reconstruction error 3.64%, the smallest reconstruction error in the test set. By comparing the
Fig. 6. Average relative errors of cellular normal detonations case test set.
reconstructed and numerical results, it is observed that the discontinuities with drastic changes in the flow field will be smoothed, because some modes are discarded in the reconstruction. Therefore, the reconstruction error of Fig. 7(b) is relatively large due to containing more areas behind the Mach stem, and vice versa. In
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Fig. 7. Typical reconstruction results of cellular normal detonations (up: original field, down: reconstructed field).
Fig. 8. Pressure and temperature fields of a typical unstable oblique detonation.
general, the detail shock structure of cellular detonation can be reproduced preciously, although there are some unavoidable errors. 3.2. Unstable oblique detonations To test the reconstruction method, cases of reconstructing the unstable oblique detonations with a dimensionless activation energy Ea = 30 are examined further. Oblique detonation wave may have the smooth and two types of unstable surfaces, one is featured by left-running triple points and the other is featured by two groups of triple points, both left-running and right-running [42–44]. These unstable surfaces are different from those in normal detonations, so we use them to examine whether the proposed method is general to reconstruct various cellular surfaces. First of all, unstable oblique detonation flow fields at different instants were simulated using the numerical methods introduced in Appendixes. The typical transient temperature and pressure field are shown in Fig. 8. It is observed that the oblique shock to oblique detonation wave transition occurs around x = 20, and the smooth surface of oblique detonation wave appears after the initiation. Nevertheless, the oblique detonation wave surface becomes cellular due to the formation of triple points and the transverse waves. A data set consisting of 400 transient oblique detonation wave surface flow fields is extracted after the general oblique detonation structure reaches a quasi-steady state. The flow field data are taken from a rectangular region, which is parallel to the oblique detonation wave surface and located at x = 80.5 and y = 12.5 with a height of 5.5 and a length of 6.5, respectively. The data set is divided ran-
Fig. 9. The first six POD modes of unstable oblique detonation wave.
domly as the training set and validation set, with the ratio 85% and 15%, and additional 50 different transient oblique detonation flow fields are further generated from other transient flow fields as test set. The reconstructed region has the height of 5.5, and in the reconstruction, the height of input sub-field is chosen to be 3.67, 2/3 of the full field. The first six POD modes of the 400 transient full fields of oblique detonation wave sequence are shown in Fig. 9. Similar to the POD modes of cellular normal detonations, the first mode
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Fig. 10. POD approximations with different reconstruction precision.
Table 2 ANN parameters for the oblique detonations case.
unstable
Layers
Neurons number
Input layer Hidden layer 1 Hidden layer 2 Output layer
32 25 40 55
is the main mode representing the averaged macroscope flow field structure of the oblique detonation wave. Compared with the rough surface of cellular normal detonations Mode 1 in Fig. 4, the averaged macro-wave surface of unstable oblique detonations Mode 1 is seen to be flatter. Since oblique detonation wave is an overdriven detonation wave, whose wave surface cellular structure evolution is restricted by high-speed inflow to a certain extent. Modes 2–6 show the different local small-scale wave structures of the cellular surface. The approximate solutions with different POD modes preserved for typical unstable oblique detonation full fields are shown in Fig. 10, by which the ANN output layer neurons number is determined to be 55. Some other prescribed ANN parameters are also given in Table 2. The average relative errors of unstable oblique detonations case test set are plotted in Fig. 11. Unfortunately, the average approximation error of the test set is 24.81%, higher than 9.39% of cellular normal detonations case. Some typical reconstruction results of unstable oblique detonations are shown in Fig. 12. Among them, Fig. 12(a) corresponds to the reconstructed results of the 18th transient flow field in Fig. 11, close to the average error of the test set; Fig. 12(b) corresponds to the reconstructed results of the 2nd transient flow field in Fig. 11, corresponds to the largest reconstruction error in the test set; Fig. 12(c) corresponds to the reconstructed results of the 31st transient flow field in Fig. 11, corresponds to the smallest reconstruction error in the test set. Compared with cellular normal detonations, the reconstruction error of unstable oblique detonations surface rises obviously. This should be attributed to the regularity of cellular structures, which depends on the activation energy controlling the heat release process. For the normal detonation wave with relatively low activation energy, the cellular structure is regular, but the unstable oblique
Fig. 11. Average relative errors of unstable oblique detonations case test set.
detonation wave has more irregular surface. Then, ANN is difficult to predict the dynamic characteristics of oblique detonation wave cellular structure with fewer modes accurately. Therefore, more reconstruction modes need to be retained for high-precision reconstruction of oblique detonation wave surface, which poses a greater challenge to the designing and training of ANN. 4. Conclusion A reconstruction method combining POD and ANN is proposed to develop an alternative approach to get the detonation surface, rather not direct experimental measuring. The detonation flow field is decomposed into a set of fixed basic modes along with time-varying coefficients by means of POD, so that the flow field with a large number of parameters can be represented by a small number of coefficients. According to the required reconstruction accuracy, the coefficients are further reduced to relieve the ANN designing and training. The mapping relationship between the reduced coefficients of full flow field and those of the post-surface sub-field is established by ANN to reconstruct the detonation wave surface. The feasibility of the reconstruction method is verified by the reconstruction of the cellular surface of both cellular normal detonations and unstable oblique detonations.
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Fig. 12. Typical reconstruction results of unstable oblique detonations (up: original field, down: reconstructed field).
Only the pressure flow field is used in the reconstruction to demonstrate the ability of the reconstruction method. Theoretically, this method is independent of the used variables, so can be further extended to reconstruct the velocity field, species field, et al. Further study should implement the reconstruction based on different variables collectively, which may be critical to improve the method’s performance. It should be noted that the proposed method actually provides a universal frame for the research of detonation surface. In principle, the reconstruction can be implemented based on numerical or experimental results from the same source, so has the potential to be employed in other numerical and experimental cases. However, whether the data from different sources can be used together, such as combing numerical and experimental results, is a promising research direction. Finally, POD method is used to decompose the detonation flow fields into basic features, which also brings a new angle to describe and understand the complicated unstable detonation phenomenon. Inherent ordered structure always exists in complex flow field, fully utilizing of these essential characteristics can bring great convenience to research.
Declaration of Competing Interest We have no relevant interest(s) to disclose.
Acknowledgments The research is supported by the National Natural Science Foundation of China NSFC (Nos. 11822202 and 91641130).
Appendix A The non-dimensional governing equations with a single-step, irreversible chemical reaction model are of the form:
∂U ∂ E ∂ F + + + S = 0, ∂t ∂ x ∂ y
(A1)
⎡ρ⎤
⎡0⎤ ρv ⎤ 2 ⎢ρ u⎥ ⎢ ρu + p ⎥ ⎢ ρ uv ⎥ ⎢0⎥ U = ⎢ ρv ⎥, E = ⎢ ρ uv ⎥, F = ⎢ ρv2 + p ⎥, S = ⎢ 0 ⎥, ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ρe 0 ρ u (e + p) ρv(e + p) ρλ ω˙ ρ uλ ρvλ ⎡
ρu
⎤
⎡
(A2) with
e=
(γ
p 1 2 + u + v2 − λQ, − 1 )ρ 2
(A3)
p = ρT,
(A4)
ω˙ = kρ (1 − λ ) exp (−Ea /T ).
(A5)
All the flow variables have been made dimensionless by reference to the uniform unburned state ahead of the detonation front,
ρ=
ρ˜ p˜ T˜ u˜ v˜ Q˜ ,p= ,T = ,u = ,v = ,Q = , ˜ ˜ ρ˜0 p˜ 0 T0 RT˜0 R˜T˜0 R˜T˜0
Ea =
E˜a . R˜T˜0
(A6)
The variables ρ , u, v, p, e and Q are the density, velocities in xand y- direction, pressure, total energy, and the amount of chemical heat release, respectively. For the chemical reaction, λ is the reaction progress variable which varies between 0 (for unburned reactant) and 1 (for product). The reaction is controlled by the activation energy Ea and the pre-exponential factor k, which is chosen to define the spatial and temporal scales, so the half reaction zone length is unit. The governing equations are discretized on Cartesian uniform grids and solved numerically using the MUSCL-Hancock scheme with Strang’s splitting. The MUSCL-Hancock scheme is formally a second-order extension to Godunov’s first order upwind method by constructing the Riemann problem on the inter-cell boundary [45]. The scheme is made total variation diminishing (TVD) with the
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Fig. B.1. A schematic of an oblique detonation wave induced by the wedge in the combustible gas mixtures.
use of slope limiter MINBEE, and the Harten-Lax-van Leer-Contact (HLLC) approximate solver is used for the Riemann problem. In the simulations, we use the dimensionless parameters Q = 50 and γ = 1.2. These are used traditionally in numerical simulations as canonical values to investigate detonation wave phenomena. Appendix B On the cellular normal detonations, the data set of cellular detonation wave flow fields is extracted from the simulation results of detonation wave propagating in a rectangular tube. The slip boundary conditions are used on the upper and bottom wall of the tube, while zero-gradient boundary conditions extrapolated from the interior are implemented on the left and the right boundaries. Initially static unburned gas with unity density and pressure fulfills the whole tube. The ignition zone with high temperature and pressure is used to initiate the detonation, and self-sustained detonations with nearly Chapman-Jouguet velocity form after traveling certain distance. A long-distance simulation is performed to eliminate the effects of initiation zone. Relatively lower activation energy, Ea = 5, is used to get cellular detonation waves. Because of low Ea , about 10 grids per unit length is enough to converge and used for the following simulations. On the unstable oblique detonations, a schematic of the wedgeinduced oblique detonation is shown in Fig. B.1. The combustible supersonic inflow with incident Mach number M0 reflects on the two-dimensional wedge with angle θ , and high temperature behind the oblique shock wave (OSW) may trigger exothermic chemical reactions and lead to the onset of an oblique detonation wave (ODW). The computational domain is shown in region bounded by the dashed line, whose coordinates are aligned with the wedge surface. Initially the whole flow field has uniform density, velocity, and pressure. Both the density and pressure are unity as the unburned state, and the velocity is calculated and projected according to M0 and θ . Inflow conditions are fixed at the free-stream values in both the left and upper boundaries of the domain. Outflow conditions extrapolated from the interior are implemented on the right and lower boundaries before the wedge. Slip boundary conditions are used on the wedge surface, which starts from x = 0.5 on the lower boundary. In this study, M0 and θ are fixed at 12.5° and 26°, respectively. The activation energy of the combustible mixture Ea = 30. And 32 points per half reaction length is used to perform the oblique detonation simulation, which is enough to simulate the unstable surface of the oblique detonation [43,44]. References [1] J.H.S. Lee, The detonation phenomenon, Cambridge University Press, New York, 2008. [2] J.E. Shepherd, Detonation in gases, Proc. Combust. Inst. 32 (2009) 83–98. [3] E.S. Oran, Understanding explosions-from catastrophic accidents to creation of the universe, Proc. Combust. Inst. 35 (2015) 1–35.
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Contents lists available at ScienceDirect
Combustion and Flame journal homepage: www.elsevier.com/locate/combustflame
Reconstructing cellular surface of gaseous detonation based on artificial neural network and proper orthogonal decomposition Yining Zhang a, Lin Zhou a,b, Hao Meng a, Honghui Teng b,∗ a b
State Key Laboratory of Laser Propulsion & Application, Beijing Power Machinery Research Institute, Beijing 100074, China School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
a r t i c l e
i n f o
Article history: Received 22 August 2019 Revised 22 October 2019 Accepted 23 October 2019
Keywords: Cellular detonation Oblique detonation Proper orthogonal decomposition Artificial neural network
a b s t r a c t Gaseous detonation has complicated cellular surface, whose comprehensive investigation is critical not only to the detonation physics but also the detonation engine development. Because measuring the highresolution dynamic surface is beyond the present experimental technical skills, we propose a reconstruction method of detonation wave surface based on post-surface flow field. This method combines two technologies, the proper orthogonal decomposition (POD) in fluid research and the artificial neural network (ANN) in machine learning research. POD is employed to extract the main features of flow fields, and the pre-trained ANN builds up the connection between the reduced coefficients of full flow fields and post-surface flow fields. The reconstruction is tested through the numerical results from one-step irreversible heat release model, displaying a good performance in both cellular normal detonations and unstable oblique detonations. The method may provide a universal frame for the detonation research, and has the potential to be employed in other numerical and experimental results. © 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Gaseous detonation is an extreme combustion phenomenon and has been studied widely for its extensively existing in catastrophic explosion and astrophysics [1–3]. Recently, detonation research on its astronautic propulsion attracts more and more attention to develop high-efficiency engines [4,5]. The new trend needs underlying understanding on the wave dynamics, which provides the foundation for the engine design and adjusting the waves inside. As shock-induced combustion, the detonation dynamics is dominated by the surface composed of shock-flame complex. Different from classic ZND (Zeldovich-von Neumann-Döring) model prediction, realistic wave surfaces are cellular due to reciprocating transverse waves [6]. Hence, cellular structures become one of the most important direction in the detonation physics research, such as [7–13]. Recording tracks of triple points through smoked foils has been the widely used technique to obtain the evolution history of the detonation wave structure in past decades [14–18]. However, in the detonation engine research, the wave surface is usually treated as a smooth one following the ZND model [19,20]. Recent numerical study [21,22] demonstrated that the non-premixed inflow results in a very complicated wave system in rotating detonation en-
∗
Corresponding author. E-mail address: [email protected] (H. Teng).
gines (RDE), but very limited experimental results, such as [23,24], are available due to the difficulty of measurement technology. This is because the highly unstable surface, deriving from cellular shock-flame complex, brings much uncertainty on the quantitative measurement. To overwhelming these difficulties, an alternative approach is measuring the post-surface flow field directly, and reconstructing the surface based on the measured information and machine learning technique. Since the post-surface region has relatively large time and length scales, this indirect method has much lower technical challenge than measuring wave surface directly. A reconstruction method of detonation wave surface based on the post-surface flow field is proposed in this study. The key idea is to establish a connection between full flow field of gaseous detonation and corresponding post-surface flow field using an artificial neural network (ANN). As an extensively used data-driven technique, ANN has shown its powerful ability to learning the underlying mechanism from training by large data set [25–28]. Considering the experimental results are limited and as the first step, the numerical results from one-step irreversible heat release model are used here to check the feasibility and problems of the proposed method. To be used in ANN, the features of shock-flame complex should be extracted first, which has been performed through proper orthogonal decomposition (POD). POD method is a very effective feature extraction and dimension reduction method, which has been widely used in turbulence and combustion fields [29–32]. ANN and POD methods have been used collectively in
https://doi.org/10.1016/j.combustflame.2019.10.031 0010-2180/© 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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flow filed reconstruction to relieve the curse of the dimensionality in aerodynamic [33]. However, this proposed method has the essential difference because of the reconstruction from post-surface flow field, rather not simplified problem. The method is introduced in Section 2 in detail, and two serial cases are tested in Section 3, demonstrating a good ability in both cellular normal detonations and unstable oblique detonations. The conclusion and discussion on future work are given in Section 4. 2. Reconstruction method 2.1. Introduction on proper orthogonal decomposition The reconstruction method actually builds up a mapping from post-surface flow field to the full flow field, and the mapping of this study is achieved through an ANN. However, describing the flow field requires large amounts of data, e.g., the parameters in each mesh, so the flow field cannot be used directly as the input and output of ANN. Fortunately, there are still some fixed basic modes (features) of detonation flow fields despite of the transient flow field varying at different times, so different combinations of these modes can constitute the transient flow field. The POD is a method of decomposing the transient flow field sequence into linear combinations of a set of fixed basic modes along with timevarying coefficients [34]. Thereafter, only the coefficients of these modes at certain time need to be known to reconstruct the corresponding flow field. The optimality of the POD reduces the amount of information required to represent statistically dependent data to a minimum [35]. Moreover, the POD also captures the essential dynamics of the flow and provides insight into the nature of the flow and its instabilities [36]. In conclusion, the POD here is used to compress flow field efficiently with representing flow field with fixed modes and corresponding coefficients, so favoring the ANN training. For simulated flow fields, the method of snapshots [37,38] is used to implement the POD. According to the simulation grid, the transient flow field can be arranged into a n-dimensional vector si in rows or columns order. Then the si at m different instants can be arranged in a sequence to form a temporal snapshots matrix S = {s1 , s2 , . . ., sm }. Each snapshot si contains a quantity of scalar flow magnitudes which in the case of experimental results usually represents the velocity information (PIV (particle image velocimetry) technology). In the case of numerical simulations these snapshots may represent other useful information such as pressure, temperature, species, etc. In this paper, pressure field extracted from the simulation results is used as snapshot si to represent the wave structure of detonation flow field. Then, we compute the reduced singular value decomposition (SVD) of snapshots matrix S:
S = U V
T
(1)
Here, U is an n × r matrix whose columns consist of the POD modes vectors uj . is an r × r diagonal matrix whose diagonal components are the singular values σ j of S. They describe the energy contained in each mode which indicating the relevance of each mode in the total flow field [39]. V is an m × r matrix and each component of the row vector ci of VT describes the contribution of each mode uj to the snapshot flow field si . Thus, si can be represented as a linear combination of fixed basic modes uj according to the coefficient vector ci with k = r:
si =
k
u j ci j
(2)
j=1
To further relieve the ANN designing and training, si can be approximated following Eq. (2) but with a reduced k < r number of
Fig. 1. Basic architecture of multiply feedforward ANN.
modes. The number of retained modes k used to reconstruct the full flow field depends on the reconstruction accuracy. 2.2. Introduction on artificial neural network After the flow fields are decomposed into the fixed modes and corresponding coefficients by POD, the next step is using ANN to establish the mapping from the reduced modes coefficients of post-surface sub-field to those of the full flow field. The basic architecture of typical multiply feedforward ANN is shown in Fig. 1. The network includes one input layer, several hidden layers, and one output layer. In each layer, there are several neurons which connect with neurons of other layers. Unfortunately, there are no universal and effective rules to determine the detailed number of hidden layers and neurons in ANN research fields. In this work, we use two hidden layers, whose neurons are varied depending on the reconstruction problems. The Nguyen-Widrow layer initialization function is used as the ANN initial method [40], and weight and bias values of neurons are updated according to the scaled conjugate gradient algorithm [41]. The scaled conjugate gradient algorithm is based on conjugate directions, but a step size scaling mechanism is used to avoid a time-consuming line-search per learning iteration, making the algorithm achieve better superlinear convergence rate. The performance function used to evaluate the ANN is mean-squared error along with the regularization term, which is used to relieve overfitting. Moreover, validation is used to stop training early if the network performance on the validation set fails to improve or remains the same for certain pre-set epochs. 2.3. Algorithm of flow field reconstruction By combining the POD and ANN, the full field can be reconstructed from the post-surface sub-field. A flowchart illustrating the present algorithm of flow field reconstruction is shown in Fig. 2. The full flow fields and corresponding sub-fields snapshots set are extracted first from the transient flow fields based on the simulation results to carry out the training of ANN. It is worth noting that, to ensure the reconstruction accuracy, the POD modes of sub-fields and full flow fields from the training and validation set should be basically consistent to those from the test set. That is, it’s necessary that the detonation wave can complete several cycles of change process during the time period of extracting transient flow field. Thus, the training set can contain typical dynamic features of transient wave structure as much as possible. Then, decomposing both the full fields and sub-fields of training and validation set into basic modes and corresponding coefficients set through POD according to Eq. (1). Thereafter, determining the number of full flow field modes need to be retained to satisfy the required accuracy of reconstruction, which corresponding to
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Fig. 2. Flowchart of the wave surface reconstruction algorithm based on ANN and POD.
the number of neurons No in the output layer of ANN. As for the input layer neurons number Ni , generally speaking, the more subfield information provided to ANN input, the more accurate the reconstructed result will be. On the other side, too much input will lead to the training of ANN difficult. Since the sub-fields usually have relatively slow time- and space-varying flow field structure, the corresponding retained POD modes, i.e., the input layer neurons number Ni , are determined to be less than those of the full fields, i.e., the output layer neurons number No . In this work, we conduct several trial calculations to determine the detailed number of neurons in the input layer and two hidden layers of ANN, which are Ni , Nl1 and Nl2 , respectively. In practice, we gradually increase the number of neurons and do various trainings, the network with the smallest error is chosen to be the final model. Furthermore, the number of neurons in the hidden layers are chosen to be close to the input and output layer neurons number, which can be helpful to construct a simple ANN. Finally, we train the ANN with both training and validation set to establish the mapping relationship between the first Ni coefficients corresponding to the post-surface sub-fields modes and the first N° coefficients corresponding to full flow fields modes. Once the time-consuming training finished, reconstructing the full flow field using the trained ANN requires only simple and fast algebraic computation. For any post-surface sub-field A in the test set, the corresponding sub-field modes coefficients can be easily calculated by using the known sub-fields POD modes. Then, the well-trained ANN is used to compute the coefficients corresponding to the full flow field modes of A. Finally, the reconstructed full flow field A can be obtained by the linear superposition of the known full flow field POD modes according to Eq. (2). In order to evaluate the reconstruction performance quantitatively, the average relative error between the reconstructed full field and the real full field is defined as follows:
Error =
M N 1 P˜m,n − Pm,n Pm,n × 100% MN
(3)
m=1 n=1
Here Pm,n and P˜m,n denote the real and reconstructed pressure in the grid point (m, n), respectively. M and N denote the number of reconstructed flow field points in the x- and y- direction.
3. Results and discussion 3.1. Cellular normal detonations The proposed method is first used to reconstruct the cellular surface of normal detonation propagating in a rectangular tube. First of all, cellular detonation flow fields at different instants, like those shown in Fig. 3, were simulated using the numerical methods introduced in Appendixes. Then, a data set consisting of 300 transient detonation flow fields is extracted after the general detonation structure reaches a quasi-steady state. The data set is divided randomly as the training set and validation set, with the ratio 85% and 15%, and additional 50 different transient detonation flow fields are further generated from other transient flow fields as test set. The extracted region has the height of 12, with the leading shock kept near 10. In the training and test processes, the sub-field has the same width with full field, while the height is chosen to be 2/3 of the full field, that is 8 from the bottom. Application of the POD to the 300 transient cellular detonation full field snapshots yields a set of modes with each mode indicating some inner features of the flow field. The first six POD modes are shown in Fig. 4. The first mode is the main mode which stands for the averaged macroscope leading shock wave structure of the cellular detonation wave surface. Mode 2 to 6 show the different shock-flame complex composed of leading shock and transverse waves. Different combinations of all modes can exhibit the complex dynamic process of the cellular detonation wave. To determine how many modes are needed to reconstruction full flow field, we project typical flow fields onto the different retained modes, and use Eq. (2) to recover the approximated solutions, shown in Fig. 5. As expected, the approximation to the CFD solution improves with increasing retained modes of reconstruction, and 30 modes retained is able to generate a good approximation. Thus, the ANN output layer neurons number is set to be 30, and other ANN parameters used in the cellular normal detonations case are shown in Table 1. Using the well-trained ANN, we reconstruct the full field from the post-surface sub-field, and a test case including totally 50 flow fields chosen randomly from simulation results is examined to evaluate the method’s performance. The average relative errors calculated by Eq. (3) are shown in Fig. 6. Obviously, more than half of the reconstruction errors are smaller than 10%, and the average reconstruction error of this test set is about 9.39%, which is accept-
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Fig. 3. Cellular normal detonation wave surface at different moments.
Fig. 4. The first six POD modes of cellular normal detonation wave.
Fig. 5. POD approximations with different reconstruction precision.
Table 1 ANN parameters for the cellular normal detonations case. Layers
Neurons number
Input layer Hidden layer 1 Hidden layer 2 Output layer
24 25 35 30
able considering several parameters can be optimized in the future work. To analyze the reconstruction results, some typical flow fields are shown in Fig. 7. The first one in Fig. 7(a) corresponds to the 11th transient flow field in Fig. 6, with the relative reconstruction error 9.45%, close to the average error of the test set. It can be seen that the triple points and transverse waves are reproduced clearly, except that the wave surface has a little bit blurred without affecting the distinguished cellular structure. The reconstructed flow field with the largest reconstruction error 17.42% in the test set is shown in Fig. 7(b), corresponding to the 4th transient flow field in Fig. 6. In contrast, Fig. 7(c) corresponds to the 23rd transient flow field in Fig. 6, with the relative reconstruction error 3.64%, the smallest reconstruction error in the test set. By comparing the
Fig. 6. Average relative errors of cellular normal detonations case test set.
reconstructed and numerical results, it is observed that the discontinuities with drastic changes in the flow field will be smoothed, because some modes are discarded in the reconstruction. Therefore, the reconstruction error of Fig. 7(b) is relatively large due to containing more areas behind the Mach stem, and vice versa. In
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Fig. 7. Typical reconstruction results of cellular normal detonations (up: original field, down: reconstructed field).
Fig. 8. Pressure and temperature fields of a typical unstable oblique detonation.
general, the detail shock structure of cellular detonation can be reproduced preciously, although there are some unavoidable errors. 3.2. Unstable oblique detonations To test the reconstruction method, cases of reconstructing the unstable oblique detonations with a dimensionless activation energy Ea = 30 are examined further. Oblique detonation wave may have the smooth and two types of unstable surfaces, one is featured by left-running triple points and the other is featured by two groups of triple points, both left-running and right-running [42–44]. These unstable surfaces are different from those in normal detonations, so we use them to examine whether the proposed method is general to reconstruct various cellular surfaces. First of all, unstable oblique detonation flow fields at different instants were simulated using the numerical methods introduced in Appendixes. The typical transient temperature and pressure field are shown in Fig. 8. It is observed that the oblique shock to oblique detonation wave transition occurs around x = 20, and the smooth surface of oblique detonation wave appears after the initiation. Nevertheless, the oblique detonation wave surface becomes cellular due to the formation of triple points and the transverse waves. A data set consisting of 400 transient oblique detonation wave surface flow fields is extracted after the general oblique detonation structure reaches a quasi-steady state. The flow field data are taken from a rectangular region, which is parallel to the oblique detonation wave surface and located at x = 80.5 and y = 12.5 with a height of 5.5 and a length of 6.5, respectively. The data set is divided ran-
Fig. 9. The first six POD modes of unstable oblique detonation wave.
domly as the training set and validation set, with the ratio 85% and 15%, and additional 50 different transient oblique detonation flow fields are further generated from other transient flow fields as test set. The reconstructed region has the height of 5.5, and in the reconstruction, the height of input sub-field is chosen to be 3.67, 2/3 of the full field. The first six POD modes of the 400 transient full fields of oblique detonation wave sequence are shown in Fig. 9. Similar to the POD modes of cellular normal detonations, the first mode
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Fig. 10. POD approximations with different reconstruction precision.
Table 2 ANN parameters for the oblique detonations case.
unstable
Layers
Neurons number
Input layer Hidden layer 1 Hidden layer 2 Output layer
32 25 40 55
is the main mode representing the averaged macroscope flow field structure of the oblique detonation wave. Compared with the rough surface of cellular normal detonations Mode 1 in Fig. 4, the averaged macro-wave surface of unstable oblique detonations Mode 1 is seen to be flatter. Since oblique detonation wave is an overdriven detonation wave, whose wave surface cellular structure evolution is restricted by high-speed inflow to a certain extent. Modes 2–6 show the different local small-scale wave structures of the cellular surface. The approximate solutions with different POD modes preserved for typical unstable oblique detonation full fields are shown in Fig. 10, by which the ANN output layer neurons number is determined to be 55. Some other prescribed ANN parameters are also given in Table 2. The average relative errors of unstable oblique detonations case test set are plotted in Fig. 11. Unfortunately, the average approximation error of the test set is 24.81%, higher than 9.39% of cellular normal detonations case. Some typical reconstruction results of unstable oblique detonations are shown in Fig. 12. Among them, Fig. 12(a) corresponds to the reconstructed results of the 18th transient flow field in Fig. 11, close to the average error of the test set; Fig. 12(b) corresponds to the reconstructed results of the 2nd transient flow field in Fig. 11, corresponds to the largest reconstruction error in the test set; Fig. 12(c) corresponds to the reconstructed results of the 31st transient flow field in Fig. 11, corresponds to the smallest reconstruction error in the test set. Compared with cellular normal detonations, the reconstruction error of unstable oblique detonations surface rises obviously. This should be attributed to the regularity of cellular structures, which depends on the activation energy controlling the heat release process. For the normal detonation wave with relatively low activation energy, the cellular structure is regular, but the unstable oblique
Fig. 11. Average relative errors of unstable oblique detonations case test set.
detonation wave has more irregular surface. Then, ANN is difficult to predict the dynamic characteristics of oblique detonation wave cellular structure with fewer modes accurately. Therefore, more reconstruction modes need to be retained for high-precision reconstruction of oblique detonation wave surface, which poses a greater challenge to the designing and training of ANN. 4. Conclusion A reconstruction method combining POD and ANN is proposed to develop an alternative approach to get the detonation surface, rather not direct experimental measuring. The detonation flow field is decomposed into a set of fixed basic modes along with time-varying coefficients by means of POD, so that the flow field with a large number of parameters can be represented by a small number of coefficients. According to the required reconstruction accuracy, the coefficients are further reduced to relieve the ANN designing and training. The mapping relationship between the reduced coefficients of full flow field and those of the post-surface sub-field is established by ANN to reconstruct the detonation wave surface. The feasibility of the reconstruction method is verified by the reconstruction of the cellular surface of both cellular normal detonations and unstable oblique detonations.
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Fig. 12. Typical reconstruction results of unstable oblique detonations (up: original field, down: reconstructed field).
Only the pressure flow field is used in the reconstruction to demonstrate the ability of the reconstruction method. Theoretically, this method is independent of the used variables, so can be further extended to reconstruct the velocity field, species field, et al. Further study should implement the reconstruction based on different variables collectively, which may be critical to improve the method’s performance. It should be noted that the proposed method actually provides a universal frame for the research of detonation surface. In principle, the reconstruction can be implemented based on numerical or experimental results from the same source, so has the potential to be employed in other numerical and experimental cases. However, whether the data from different sources can be used together, such as combing numerical and experimental results, is a promising research direction. Finally, POD method is used to decompose the detonation flow fields into basic features, which also brings a new angle to describe and understand the complicated unstable detonation phenomenon. Inherent ordered structure always exists in complex flow field, fully utilizing of these essential characteristics can bring great convenience to research.
Declaration of Competing Interest We have no relevant interest(s) to disclose.
Acknowledgments The research is supported by the National Natural Science Foundation of China NSFC (Nos. 11822202 and 91641130).
Appendix A The non-dimensional governing equations with a single-step, irreversible chemical reaction model are of the form:
∂U ∂ E ∂ F + + + S = 0, ∂t ∂ x ∂ y
(A1)
⎡ρ⎤
⎡0⎤ ρv ⎤ 2 ⎢ρ u⎥ ⎢ ρu + p ⎥ ⎢ ρ uv ⎥ ⎢0⎥ U = ⎢ ρv ⎥, E = ⎢ ρ uv ⎥, F = ⎢ ρv2 + p ⎥, S = ⎢ 0 ⎥, ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ρe 0 ρ u (e + p) ρv(e + p) ρλ ω˙ ρ uλ ρvλ ⎡
ρu
⎤
⎡
(A2) with
e=
(γ
p 1 2 + u + v2 − λQ, − 1 )ρ 2
(A3)
p = ρT,
(A4)
ω˙ = kρ (1 − λ ) exp (−Ea /T ).
(A5)
All the flow variables have been made dimensionless by reference to the uniform unburned state ahead of the detonation front,
ρ=
ρ˜ p˜ T˜ u˜ v˜ Q˜ ,p= ,T = ,u = ,v = ,Q = , ˜ ˜ ρ˜0 p˜ 0 T0 RT˜0 R˜T˜0 R˜T˜0
Ea =
E˜a . R˜T˜0
(A6)
The variables ρ , u, v, p, e and Q are the density, velocities in xand y- direction, pressure, total energy, and the amount of chemical heat release, respectively. For the chemical reaction, λ is the reaction progress variable which varies between 0 (for unburned reactant) and 1 (for product). The reaction is controlled by the activation energy Ea and the pre-exponential factor k, which is chosen to define the spatial and temporal scales, so the half reaction zone length is unit. The governing equations are discretized on Cartesian uniform grids and solved numerically using the MUSCL-Hancock scheme with Strang’s splitting. The MUSCL-Hancock scheme is formally a second-order extension to Godunov’s first order upwind method by constructing the Riemann problem on the inter-cell boundary [45]. The scheme is made total variation diminishing (TVD) with the
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Fig. B.1. A schematic of an oblique detonation wave induced by the wedge in the combustible gas mixtures.
use of slope limiter MINBEE, and the Harten-Lax-van Leer-Contact (HLLC) approximate solver is used for the Riemann problem. In the simulations, we use the dimensionless parameters Q = 50 and γ = 1.2. These are used traditionally in numerical simulations as canonical values to investigate detonation wave phenomena. Appendix B On the cellular normal detonations, the data set of cellular detonation wave flow fields is extracted from the simulation results of detonation wave propagating in a rectangular tube. The slip boundary conditions are used on the upper and bottom wall of the tube, while zero-gradient boundary conditions extrapolated from the interior are implemented on the left and the right boundaries. Initially static unburned gas with unity density and pressure fulfills the whole tube. The ignition zone with high temperature and pressure is used to initiate the detonation, and self-sustained detonations with nearly Chapman-Jouguet velocity form after traveling certain distance. A long-distance simulation is performed to eliminate the effects of initiation zone. Relatively lower activation energy, Ea = 5, is used to get cellular detonation waves. Because of low Ea , about 10 grids per unit length is enough to converge and used for the following simulations. On the unstable oblique detonations, a schematic of the wedgeinduced oblique detonation is shown in Fig. B.1. The combustible supersonic inflow with incident Mach number M0 reflects on the two-dimensional wedge with angle θ , and high temperature behind the oblique shock wave (OSW) may trigger exothermic chemical reactions and lead to the onset of an oblique detonation wave (ODW). The computational domain is shown in region bounded by the dashed line, whose coordinates are aligned with the wedge surface. Initially the whole flow field has uniform density, velocity, and pressure. Both the density and pressure are unity as the unburned state, and the velocity is calculated and projected according to M0 and θ . Inflow conditions are fixed at the free-stream values in both the left and upper boundaries of the domain. Outflow conditions extrapolated from the interior are implemented on the right and lower boundaries before the wedge. Slip boundary conditions are used on the wedge surface, which starts from x = 0.5 on the lower boundary. In this study, M0 and θ are fixed at 12.5° and 26°, respectively. The activation energy of the combustible mixture Ea = 30. And 32 points per half reaction length is used to perform the oblique detonation simulation, which is enough to simulate the unstable surface of the oblique detonation [43,44]. References [1] J.H.S. Lee, The detonation phenomenon, Cambridge University Press, New York, 2008. [2] J.E. Shepherd, Detonation in gases, Proc. Combust. Inst. 32 (2009) 83–98. [3] E.S. Oran, Understanding explosions-from catastrophic accidents to creation of the universe, Proc. Combust. Inst. 35 (2015) 1–35.
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