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Citation: Ambrutis, A.; Povilaitis, M.

Development of a CFD-Suitable Deep

Neural Network Model for Laminar

Burning Velocity. Appl. Sci. 2022,12,

7460. https://doi.org/10.3390/

app12157460

Academic Editor: Talal Yusaf

Received: 30 June 2022

Accepted: 23 July 2022

Published: 25 July 2022

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applied

sciences

Article

Development of a CFD-Suitable Deep Neural Network Model

for Laminar Burning Velocity

Andrius Ambrutis * and Mantas Povilaitis

Laboratory of Nuclear Installation Safety, Lithuanian Energy Institute, Breslaujos g. 3, LT-44403 Kaunas,

Lithuania; mantas.povilaitis@lei.lt

*Correspondence: andrius.ambrutis@lei.lt

Featured Application: The presented DNN model has been developed for application in the CFD

simulations requiring a simpliﬁed estimation of laminar burning velocity in dry hydrogen–air

mixtures. A pair of examples using the progress variable approach, turbulent ﬂame speed closure

model and open-source CFD solver ﬂameFoam are presented at the end of the paper.

Abstract:

Hydrogen is a valued resource for today’s industry. As a fuel, it produces large amounts of

energy and creates water during the process, unlike most other polluting energy sources. However,

the safe use of hydrogen requires reliable tools able to accurately predict combustion. This study

presents the implementation of a deep neural network of laminar burning velocity of hydrogen

into an open-source CFD solver ﬂameFoam. DNN was developed based on a previously created

larger DNN, which was too large for CFD applications since the calculations took around 40 times

longer compared to the Malet correlation. Therefore, based on the original model, a faster, but still

accurate, DNN was developed and implemented into ﬂameFoam starting with version 0.10. The

paper presents the adaptation of the original DNN into a CFD-applicable version and the initial test

results of the CFD–DNN simulation.

Keywords:

turbulent premixed combustion; hydrogen; artiﬁcial neural network; CFD; laminar

burning velocity

1. Introduction

The ﬁrst concepts of non-biological brains were analyzed during World War II by

McCulloch and Pitts [

1

]. Since then, multilayer deep neural networks (DNNs) have been

widely used for numerous research and practical applications. In machine learning, the

algorithm learns from the data, using various transformations of inputs, without a user

or programmer giving it explicit instructions [

2

]. Various other machine learning (ML)

algorithms can also be used as a replacement for neural networks since studies show

that the performance of all ML algorithms depends greatly on the problem itself [

3

,

4

]. In

general, an artiﬁcial neural network (ANN) performs better in ﬁnding complex behavior

and patterns in large amounts of data. As stated by studies, multi-layer ANNs, such as

deep neural networks (DNN) can reproduce any function with arbitrary precision [5,6].

Simulation of turbulent premixed combustion is challenging numerically due to the

wide range of spatial and temporal scales involved in turbulence and combustion chem-

istry. Therefore, to perform practically relevant simulations, various simpliﬁed modeling

approaches are usually employed, e.g., RANS treatment of turbulence or simpliﬁed com-

bustion rate estimations instead of direct modeling of chemical kinetics. The most common

approach to avoid chemical simulation is to formulate a used combustion model in a way

that chemistry would be replaced by a simpler estimation of laminar burning velocity

(LBV), e.g., from the correlations based on the experiments.

There are several examples of ANN and other ML methods for the application in the

prediction of combustible mixture properties in the literature. In 2018, Jach et al. [

7

] created

Appl. Sci. 2022,12, 7460. https://doi.org/10.3390/app12157460 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022,12, 7460 2 of 16

three ML models—ANN, SVM and multivariable regression (MR)—for LBVs of mixtures of

air with one of seven hydrocarbons from methane up to n-heptane. The best performance

was obtained by ANN model in terms of R2, RMSE and MAE.

In 2019, Mehra et al. [

8

] were able to create DNN which could predict laminar burning

velocities of HyCONG gas blends with an R-squared value approximately equal to 0.999

and a number of neurons up to 20. Concentrations of blend constituents were used as

inputs. Authors showed that an increase in network weight number decreased the testing

error, which means that DNN with more weights should have higher accuracy. Similar

behavior was noticed for the number of epochs used for model optimization. However,

while more complex networks can show better results, they also are slower and have a

higher risk of overﬁtting.

In 2020, Pulga et al. [

9

] developed methodology to improve the accuracy of laminar

ﬂame ML simulations, starting from data preparation, model creation and ﬁnishing with

model evaluation and results interpretation. Malik et al. presented a light neural network

(as it only had up to ﬁve neurons in each layer), which could get good predictions in a

short amount of time [

10

]. Their neural network could be used in real-time and had the

mean squared error equal to 0.3023 (m/s)

2

with hydrogen–air mixtures for laminar burning

velocity predictions. In their work, 577 observations were used for the training of DNN.

Part of them had a high spread and this can lower prediction accuracy. Since the amount

of available experimental data was insufﬁcient to train the neural network, the study

generated new points (up to 7300 observations) for training. Varghese and Kumar [

11

]

analyzed syngas mixtures and created multiple linear regression model to predict their

LBVs with error <10%. Their model was derived partly from the measured velocities, and

partly from the predictions using FFCM-1 kinetic mechanism.

In 2021, Eckart et al. [

12

] compared four different machine learning algorithms pre-

dicting laminar burning velocities of hydrogen–methane mixtures. Their study showed

that the DNN model can give much better predictions than other popular algorithms

such as support vector machines (SVM) or random forest while predicting velocities for

hydrogen–methane mixtures. Correa et al. [

13

] used several ML methods to predict the

research octane numbers of several spark ignition fuels. Study evaluated multiple ML

algorithms using 10-fold cross validation, which ensured that results are less inﬂuenced

by noise and more accurately reﬂects real capabilities of these models. In this study SVM

gave best predictions out of all tested methods. vom Lehn et al. [

14

] created a quantitative

structure–property relationship (QSPR) model for the estimation of LBVs of hydrocarbons

and oxygenated hydrocarbons based on their underlying fuel structures. A set of molecular

groups as well as pressure, temperature and equivalence ratio served as input parameters

to an ANN, which has been trained based on a large database of training data for 124

different compounds.

Recently, Wan et al. [

15

] compared 16 ML algorithms for the prediction of hydrocarbon

and oxygenated fuels LBVs and found out that the Gaussian or Tree-based methods gave

high predictions with most of those models explaining over 90% of all data. Li et al. [

16

]

created ML–QSPR model to screen fuels based on their predicted properties.

In our previous work [

17

], we created a DNN for the prediction of the laminar burning

velocity of hydrogen–air mixtures with a coefﬁcient of determination approximately equal

to 0.985. However, this model was too slow to be used practically (in CFD calculations)—

the testing script could make a million predictions in around 45 s, compared to the Malet

correlation [

18

], with which predictions could be made under the same conditions in less

than a second.

For complex combustion cases, increases in observation quantity can highly improve

predictions of neural networks. However, as the network starts to learn the way to solve

the problem, this effect gets less visible. In the end, if the model overlearns on data, it

can even start to mimic the noise. Therefore, the goal should be to obtain a high enough

amount of well-balanced data. Since there are more data in the speciﬁc region, the more

trustworthy prediction can be made by the neural network.

Appl. Sci. 2022,12, 7460 3 of 16

To achieve more accurate predictions, aberrant outliers may need to be removed from

the data. As suggested by J. Heaton, this can be carried out with the so-called ‘double-D’

algorithm [

19

]. The algorithm itself has many names, however, the idea behind it is to use

a double standard deviation as a removal condition. However, it is more useful when the

database has a very large number of observations. Since the amount of experimental data

on hydrogen–air LBV in the literature is not that large, this method might not signiﬁcantly

improve predictions as the neural network might not have enough data to learn well. On

the other hand, it could be a direction for future work. Moreover, outliers have a signiﬁcant

inﬂuence on the mean value as it is one of the neural network’s measures based on which it

decides how the ﬁnal model will look like. Luckily, the model can also be evaluated using

mean absolute error (MAE) instead. The inﬂuence on a model based on MAE compared

to a model based on mean squared error (MSE) was analyzed by J. Qi [

20

]. It is much

less affected by outliers but error can be inﬂuenced by the number of observations. As

pointed out by T. Chai [

21

], each metric has its advantages and disadvantages. Due to

this, it would be wise to have the MSE as a side evaluator as well and not trust in a single

measure (MAE) blindly.

As the presented research overview shows, most of the related studies to date have

focused on the development of stand-alone ML models of combustible mixtures’ properties.

While there are works considering the implementation of such a model into a CFD code [

10

],

this has not been performed up to now. Consequentially, when developing, there is less

pressure to create lean and fast, but sufﬁciently accurate models, and their suitability is

not tested in CFD frameworks. In addition, the last, but main step, producing a working,

research or application suitable CFD–ANN code is never performed. To ﬁll this gap,

we have developed a CFD-suitable deep neural network model for hydrogen LBV and

implemented it into an actual OpenFOAM-based CFD solver ﬂameFoam.

The purpose of the current paper is to, based on the literature and our previous

work [

17

], create a fast and light, suitable for CFD applications, DNN model for LBV in dry

hydrogen–air mixtures, implement the developed model into a CFD solver, and perform

testing CFD–DNN simulations of turbulent premixed combustion.

2. Developed Deep Neural Network Model for Laminar Burning Velocity

The presented model was based on a previously developed (further referred to as

the original) large DNN [17], which was too slow for CFD application. The further model

development aimed to improve the calculation speed without a signiﬁcant accuracy loss.

The original DNN model was trained on the dataset of experimental hydrogen–air mixtures

LBV at various temperatures, pressures and hydrogen concentrations, collected from the

open literature [17].

For the training of the new DNN model, the original database was expanded to 2871

data points [

22

–

47

]. Around 33% of all of them are experimental values; however, the

number of pure experimental values was insufﬁcient for the reliable training of DNN.

Therefore, additional values were interpolated from the experimental values, by adding

points from the correlation curve produced from the experimental points of the given

experiment, close to the actual experimental points. Since the results of the newest studies

are most likely to have a higher accuracy due to technological development, they were

prioritized for the interpolation, in this way giving them more ‘weight’ in the ﬁnal database,

since more data points have been added based on the newest observations and fewer points

in the area near old observations.

As can be noticed from Figure 1, most of the data are around 1 bar pressure or

298 K temperature, because most experimental works start by taking normal conditions

and changing only a single parameter. However, the expansion of the database was

performed to make the model more reliable while predicting LBVs at higher temperatures

and higher pressures, since most of the new data are under non-normal conditions. These

changes should have made the model more reliable in an entire applicability range. The

Appl. Sci. 2022,12, 7460 4 of 16

used experimental data model is applicable in the range of 295–600 K temperature and

0.1–5 bar pressure.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 18

should have made the model more reliable in an entire applicability range. The used ex-

perimental data model is applicable in the range of 295–600 K temperature and 0.1–5 bar

pressure.

(a) (b)

Figure 1. Gathered data statistics [22–47]: (a) count against pressure and (b) count against temper-

ature.

Data were split randomly into training and testing sets with around 75% of data used

to train the model and the remaining data used for validation. Further modifications such

as outlier removal were focused on the training set, avoiding modification of the testing

set, this way trying to obtain as reliable data as possible for validation.

Four hidden layers were left in the new model (same as in the original); however, to

ensure faster calculations, the number of neurons was changed to (7,10,7,5) (Figure 2) from

(40,50,30,10) in previous work [17]. The number of neurons was reduced by trying various

combinations and balancing accuracy against prediction speed.

In general, reducing the number of neurons increased the risk that the model will not

be as robust to data as the previous model. Due to this, we had to re-train the model mul-

tiple times as changes in weights could affect the predictions. Overall, this meant that the

smaller neural network required more human monitoring during the training to lessen

accuracy loss. In addition, we increased the risk that fewer neurons will lack data to learn

in some areas; to counter this, we had to increase our database. In the end, the main draw-

back related to the reduced number of neurons was that we had to put in substantial ad-

ditional effort to obtain the model with results comparable to those of larger DNN.

Figure 1.

Gathered data statistics [

22

–

47

]: (

a

) count against pressure and (

b

) count against temperature.

Data were split randomly into training and testing sets with around 75% of data used

to train the model and the remaining data used for validation. Further modiﬁcations such

as outlier removal were focused on the training set, avoiding modiﬁcation of the testing set,

this way trying to obtain as reliable data as possible for validation.

Four hidden layers were left in the new model (same as in the original); however, to

ensure faster calculations, the number of neurons was changed to (7,10,7,5) (Figure 2) from

(40,50,30,10) in previous work [

17

]. The number of neurons was reduced by trying various

combinations and balancing accuracy against prediction speed.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 18

Figure 2. Architecture of the improved model.

Several different sets of activation functions were also tested and the best predictions

were obtained when using tanh activation function for the first (dense) hidden layer

(Equation (1)) and ReLU for the rest (Equation (2)):

𝑡𝑎𝑛ℎ𝑧𝑒𝑒

𝑒+𝑒 (1)

𝑅𝑒𝐿𝑈𝑜𝑢𝑡𝑝𝑢𝑡=𝑜𝑢𝑡𝑝𝑢𝑡,𝑖𝑓𝑜𝑢𝑡𝑝𝑢𝑡0;

0,𝑒𝑙𝑠𝑒 (2)

The original model used only ReLU functions [18]; however, the switch to tanh func-

tion in the first layer gave accuracy improvement to the new model. Further inclusions of

tanh functions did not produce significant further improvement, even though in the liter-

ature there is an example of LBV–DNN with all tanh activation functions [11]. Since tanh

function takes longer to calculate and did not give a significant advantage when used in

deeper layers, we left the ReLU activation function in those layers. Result improvement

might have been achieved since the tanh activation function managed to give primary

predictions which later were adjusted by layers with ReLU bending curve so that it would

fit data better.

However, since the number of neurons is not large, it is likely that with small laminar

burning velocities (0 m/s or close to it), the model might predict values below zero. To

avoid negative predictions, the Equation (2) was also applied to the final output function.

For model optimization RMSprop optimizer was used [48]. As studies show, it is one

of the best methods for optimization [49]. In addition, the regularization parameters were

changed compared to the original model. The first layer had Ridge (L2) regularization

parameter equal to 0.000002 and other hidden layers had Lasso (L1) regularization equal

to 0.0004. These values were selected after multiple tests with varied values. The effects of

the decrease in errors due to L1 and L2 are well documented in [50,51]. In general, they

prevent DNN from overfitting, which makes the network more stable and reliable.

To improve the reliability of the new model, MAE and MSE errors were combined

and minimized as weighted errors (equal weights were selected after several test cases)

for the training of the model as a loss function. Most of the influence was displayed by

MAE since velocity was in meters per second and lower than 1 m/s. On the other hand,

MSE helped in cases when MAE became almost constant. We believe this approach could

Figure 2. Architecture of the improved model.

In general, reducing the number of neurons increased the risk that the model will

not be as robust to data as the previous model. Due to this, we had to re-train the model

multiple times as changes in weights could affect the predictions. Overall, this meant that

Appl. Sci. 2022,12, 7460 5 of 16

the smaller neural network required more human monitoring during the training to lessen

accuracy loss. In addition, we increased the risk that fewer neurons will lack data to learn in

some areas; to counter this, we had to increase our database. In the end, the main drawback

related to the reduced number of neurons was that we had to put in substantial additional

effort to obtain the model with results comparable to those of larger DNN.

Several different sets of activation functions were also tested and the best predic-

tions were obtained when using tanh activation function for the ﬁrst (dense) hidden layer

(Equation (1)) and ReLU for the rest (Equation (2)):

tanh(z)ez−e−z

ez+e−z(1)

ReLU(out put)=output,i f output >0;

0, else (2)

The original model used only ReLU functions [

18

]; however, the switch to tanh function

in the ﬁrst layer gave accuracy improvement to the new model. Further inclusions of tanh

functions did not produce signiﬁcant further improvement, even though in the literature

there is an example of LBV–DNN with all tanh activation functions [

11

]. Since tanh function

takes longer to calculate and did not give a signiﬁcant advantage when used in deeper

layers, we left the ReLU activation function in those layers. Result improvement might

have been achieved since the tanh activation function managed to give primary predictions

which later were adjusted by layers with ReLU bending curve so that it would ﬁt data better.

However, since the number of neurons is not large, it is likely that with small laminar

burning velocities (0 m/s or close to it), the model might predict values below zero. To

avoid negative predictions, the Equation (2) was also applied to the ﬁnal output function.

For model optimization RMSprop optimizer was used [

48

]. As studies show, it is one

of the best methods for optimization [

49

]. In addition, the regularization parameters were

changed compared to the original model. The ﬁrst layer had Ridge (L2) regularization

parameter equal to 0.000002 and other hidden layers had Lasso (L1) regularization equal to

0.0004. These values were selected after multiple tests with varied values. The effects of

the decrease in errors due to L1 and L2 are well documented in [

50

,

51

]. In general, they

prevent DNN from overﬁtting, which makes the network more stable and reliable.

To improve the reliability of the new model, MAE and MSE errors were combined

and minimized as weighted errors (equal weights were selected after several test cases) for

the training of the model as a loss function. Most of the inﬂuence was displayed by MAE

since velocity was in meters per second and lower than 1 m/s. On the other hand, MSE

helped in cases when MAE became almost constant. We believe this approach could give

predictions closer to real values, as performed analysis shows that averaging MAE and

MSE indeed gives values in between for the data gathered for this research. This can be

helpful in cases with areas lacking observations. Also, it is worth noticing that in cases of

unique (single) observations, the MAE and MSE will point to the same location. A simple

explanation of this could be the idea that while MAE minimizes the model towards median

and MSE—towards mean, the combination of them would predict the value in between as

visualized in Figure 3.

To obtain predictions closer to original experimental values, the study did not modify

the testing set; however, the training set for improved DNN was adjusted by making some

outliers equal to the mean. Outliers were considered values that are different by over 10%

from the median value when results from three or more different experimental sources

under the same or similar conditions were available. These values were considered outliers

since the experimental spread of more than 10% shows higher experimental uncertainties,

and given several sources, aberrant points are the most suspect. Around 3–4% of all points

were adjusted. Yet, considering that this can detect many data points near zero, those

values were ignored. Due to the way in how the model was trained, outliers still could

have inﬂuenced the ﬁnal model. However, due to the mean value, this inﬂuence is reduced.

Appl. Sci. 2022,12, 7460 6 of 16

This allowed the creation of a model which is stable and can perform well on data sets that

have outliers.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 18

give predictions closer to real values, as performed analysis shows that averaging MAE

and MSE indeed gives values in between for the data gathered for this research. This can

be helpful in cases with areas lacking observations. Also, it is worth noticing that in cases

of unique (single) observations, the MAE and MSE will point to the same location. A sim-

ple explanation of this could be the idea that while MAE minimizes the model towards

median and MSE—towards mean, the combination of them would predict the value in

between as visualized in Figure 3.

Figure 3. Predicted loss function behavior tested by comparing DNN models with different loss

functions.

To obtain predictions closer to original experimental values, the study did not modify

the testing set; however, the training set for improved DNN was adjusted by making some

outliers equal to the mean. Outliers were considered values that are different by over 10%

from the median value when results from three or more different experimental sources

under the same or similar conditions were available. These values were considered outli-

ers since the experimental spread of more than 10% shows higher experimental uncertain-

ties, and given several sources, aberrant points are the most suspect. Around 3–4% of all

points were adjusted. Yet, considering that this can detect many data points near zero,

those values were ignored. Due to the way in how the model was trained, outliers still

could have influenced the final model. However, due to the mean value, this influence is

reduced. This allowed the creation of a model which is stable and can perform well on

data sets that have outliers.

To use this model in the CFD solver, it was exported by saving the weights 𝑤,,,

biases bi,j and performing matrix multiplication with them. Here, j denotes a layer number,

and ij denotes a weight number in the j-th layer. j can obtain values from 1 to k and ij can

obtain values from 1 to nj. Then the i-th value in the j-th hidden layer hi,j is obtained from

the Equations (3) and (4):

ℎ,

∗=𝑤,,ℎ, +𝑤,,ℎ,+⋯+𝑤,,ℎ, (3)

ℎ, =𝑎𝑐𝑡𝐹ℎ,

∗+𝑏,, (4)

where actF is an activation function given by the Equations (1) and (2) for the first and

other layers, respectively. By applying Equations (3) and (4), the model can be expressed

and implemented using weights (Wj) and biases (Bj) matrixes and inputs vector (I).

Figure 3.

Predicted loss function behavior tested by comparing DNN models with different loss functions.

To use this model in the CFD solver, it was exported by saving the weights

wij−1,ij,j

,

biases b

i,j

and performing matrix multiplication with them. Here, jdenotes a layer number,

and i

j

denotes a weight number in the j-th layer. jcan obtain values from 1 to kand i

j

can

obtain values from 1 to n

j

. Then the i-th value in the j-th hidden layer h

i,j

is obtained from

the Equations (3) and (4):

h∗

i,j=wi,0,jh0,j−1+wi,1, jh1,j−1+. . . +wi,nj−1,jhnj−1,j−1(3)

hi,j=actFh∗

i,j+bi,j, (4)

where actF is an activation function given by the Equations (1) and (2) for the ﬁrst and other

layers, respectively. By applying Equations (3) and (4), the model can be expressed and

implemented using weights (W

j

) and biases (B

j

) matrixes and inputs vector (I). Weights

and biases can also be expressed as follows by Equations (5) and (6) in which nand kmark

the number of weights in interacting layers.

Wj=

w1,1 . . . wnj,1

. . . . . . . . .

w1,nj−1. . . wnj,nj−1

(5)

Bj=b1, .., bnj(6)

The output becomes the new input for the next layer. This process is repeated until all

layers are calculated and the ﬁnal output—value of LBV is predicted.

3. Comparison with Other Methods

To test the quality of the developed DNN model and to check whether some simpler

method could be sufﬁcient, other models using popular machine learning methods were

created based on the same database. All models were optimized to make a million predic-

tions in under 2 s with a testing script. Final models were tested and compared with DNN

in terms of prediction density and the most widely used metrics such as MAE, MSE and

Appl. Sci. 2022,12, 7460 7 of 16

R-squared. While density estimates are not the most optimal method of comparison, we

reduced the Gaussian kernel and estimation for all methods which was carried out with

the same kernel; therefore, we consider it to be sufﬁcient for comparison purposes. For

all models, the same training and testing datasets were used as for the developed DNN

model (75%/25%).

Density plots in Figure 4explain the high spread (Table 1) of some models (RF, MARS).

While DNN and k-NN both show good agreement with the density of data until LBV

reaches 350 cm/s, k-NN shows worse results at higher velocities. Even more, only SVM

and DNN models predict higher than 700 cm/s laminar burning velocities. On the other

hand, evaluation of SVM shows that this model is most likely to make predictions too high

at over 280 cm/s and otherwise, guesses too low. As stated by this study earlier, DNN

was trained by minimizing MAE and MSE at the same time, which should give optimal

prediction between median and mean values. It means that the model might not have the

lowest MSE and MAE values; therefore, it would be wise to evaluate it based on how well it

explained the data. However, the results show (see Table 1) that DNN has the lowest MEA

and MSE values of all tested methods with the highest being R-Squared at approximately

0.997. The second place would go to SVM regression which shows less spread than other

models (RF, k-NN, MARS). The remaining solutions performed well by showing the ability

to explain most of the data; however, their main drawback is the limited ability to predict

higher burning velocities.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 18

Figure 4. Prediction densities by multiple machine learning algorithms.

Table 1. Comparison of machine learning algorithms (values calculated against testing set).

Model MSE, (cm/s2) MAE, (cm/s) R-Squared

DNN 73.07057 6.094699 0.9972631

SVM (SVR) 843.9467 23.36927 0.9687515

Random Forest 1100.299 23.68055 0.9650517

K-Nearest Neighbors (k-NN) 1164.565 18.31963 0.9541389

MARS 1324.741 24.40525 0.9487737

The good performance of the developed DNN model can be visible in Figure 5 below,

which shows the comparison of experimental and predicted LBV values. The red line,

which shows linear regression of DNN predictions plotted against reference velocities,

almost perfectly covers the perfect match line (light blue).

Figure 4. Prediction densities by multiple machine learning algorithms.

Table 1. Comparison of machine learning algorithms (values calculated against testing set).

Model MSE, (cm/s2)MAE, (cm/s) R-Squared

DNN 73.07057 6.094699 0.9972631

SVM (SVR) 843.9467 23.36927 0.9687515

Random Forest 1100.299 23.68055 0.9650517

K-Nearest Neighbors (k-NN) 1164.565 18.31963 0.9541389

MARS 1324.741 24.40525 0.9487737

The good performance of the developed DNN model can be visible in Figure 5below,

which shows the comparison of experimental and predicted LBV values. The red line,

Appl. Sci. 2022,12, 7460 8 of 16

which shows linear regression of DNN predictions plotted against reference velocities,

almost perfectly covers the perfect match line (light blue).

Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 18

Figure 5. DNN predictions versus experimental data.

LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [18] investi-

gated the burning behavior of hydrogen mixtures and offered an expression to calculate

LBV at low equivalence ratios. A comparison of the DNN model and Malet formula pre-

dictions (Figure 6) shows that the DNN model gives predictions closer to reality with

MAE equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.

Figure 6. Malet and DNN comparison.

Figure 5. DNN predictions versus experimental data.

LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [

18

] investi-

gated the burning behavior of hydrogen mixtures and offered an expression to calculate

LBV at low equivalence ratios. A comparison of the DNN model and Malet formula predic-

tions (Figure 6) shows that the DNN model gives predictions closer to reality with MAE

equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 18

Figure 5. DNN predictions versus experimental data.

LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [18] investi-

gated the burning behavior of hydrogen mixtures and offered an expression to calculate

LBV at low equivalence ratios. A comparison of the DNN model and Malet formula pre-

dictions (Figure 6) shows that the DNN model gives predictions closer to reality with

MAE equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.

Figure 6. Malet and DNN comparison.

Figure 6. Malet and DNN comparison.

Appl. Sci. 2022,12, 7460 9 of 16

4. CFD Simulations

This chapter presents a couple of ﬂameFoam-DNN calculation examples showing that

the developed DNN model is usable with CFD simulation and provides results in line with

the Malet correlation or experimental data.

The developed model was implemented in the open-source turbulent premixed com-

bustion solver ﬂameFoam [

52

] v0.10. ﬂameFoam is an OpenFOAM-based solver using

a progress variable approach and turbulent ﬂame speed closure model to simulate com-

bustion. The turbulent ﬂame speed closure model requires turbulent ﬂame speed values,

which in turn require LBV values. In the ﬂameFoam-DNN simulations, presented be-

low, these LBV values were estimated on-the-ﬂy in the ﬂameFoam using the DNN model

programmed according to Equations (3)–(6) with the weights exported from the DNN

training. The solver source code, including the DNN model, is available on GitHub:

https://github.com/ﬂameFoam/ﬂameFoam (accessed on 29 June 2022).

To compare ﬂameFoam calculation results obtained using the DNN model versus

Malet correlation, simulation of ENACCEF2 facility (France, CNRS-ICARE) TEST1 from

ETSON-MITHYGENE benchmark [53] was performed.

ENACCEF2 is a closed vertical steel tube of 7.65 m height and 0.23 m inner diameter

(Figure 7a). Flame acceleration is achieved in the simulated experiment by 9.2 mm thin

annular obstacles situated in the lower part of the facility (Figure 7b). The ﬂame is ignited

at the bottom center and propagates upwards.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 18

4. CFD Simulations

This chapter presents a couple of flameFoam-DNN calculation examples showing

that the developed DNN model is usable with CFD simulation and provides results in

line with the Malet correlation or experimental data.

The developed model was implemented in the open-source turbulent premixed com-

bustion solver flameFoam [52] v0.10. flameFoam is an OpenFOAM-based solver using a

progress variable approach and turbulent flame speed closure model to simulate combus-

tion. The turbulent flame speed closure model requires turbulent flame speed values,

which in turn require LBV values. In the flameFoam-DNN simulations, presented below,

these LBV values were estimated on-the-fly in the flameFoam using the DNN model pro-

grammed according to Equations (3)–(6) with the weights exported from the DNN train-

ing. The solver source code, including the DNN model, is available on GitHub:

https://github.com/flameFoam/flameFoam (accessed on 29 June 2022).

To compare flameFoam calculation results obtained using the DNN model versus

Malet correlation, simulation of ENACCEF2 facility (France, CNRS-ICARE) TEST1 from

ETSON-MITHYGENE benchmark [53] was performed.

ENACCEF2 is a closed vertical steel tube of 7.65 m height and 0.23 m inner diameter

(Figure 7a). Flame acceleration is achieved in the simulated experiment by 9.2 mm thin

annular obstacles situated in the lower part of the facility (Figure 7b). The flame is ignited

at the bottom center and propagates upwards.

(a) (b) (c)

Figure 7. ENACCEF2 facility and computational grid: (a) overall scheme, (b) obstacle region close-

up and (c) computational mesh around the obstacle.

The hydrogen concentration in the homogenous combustible mixture with air was

13%. The experiment was performed at 23 °C temperature and 100,000 Pa pressure.

Figure 7.

ENACCEF2 facility and computational grid: (

a

) overall scheme, (

b

) obstacle region close-up

and (c) computational mesh around the obstacle.

The hydrogen concentration in the homogenous combustible mixture with air was

13%. The experiment was performed at 23 ◦C temperature and 100,000 Pa pressure.

The computational mesh (Figure 7c) of the facility was composed of the structured

orthogonal grids of the facility-free volume (ﬂuid region, blue in Figure 7) and facility wall

Appl. Sci. 2022,12, 7460 10 of 16

(solid region, yellow in Figure 7). 2D axisymmetric mesh with 1 mm cell sizes was used in

the presented simulations. In the ﬂuid region, ﬂameFoam solves Navier-Stokes, turbulence

and combustion equations, while in the solid region heat conductivity equation is solved.

The regions are coupled through standard OpenFOAM inter-region boundary condition

for temperature compressible::turbulentTemperatureCoupledBAfﬂeMixed.

Initial conditions were set according to the experiment. Turbulence was modeled using

the k-

ε

RANS model. Negligible values for initial turbulence parameters were selected. The

boundary condition for temperature on the external side of the solid grid was set to 23

◦

C.

Figure 8presents the ﬂame propagation velocity proﬁle (obtained from ﬂame arrival

times) obtained by both simulations. Results are very similar, showing that the DNN

model hasn’t introduced any regressions compared to the Malet correlation around the

experimental conditions [

52

]. There was no signiﬁcant difference of computation time

between Malet and DNN calculations.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 18

The computational mesh (Figure 7c) of the facility was composed of the structured

orthogonal grids of the facility-free volume (fluid region, blue in Figure 7) and facility wall

(solid region, yellow in Figure 7). 2D axisymmetric mesh with 1 mm cell sizes was used

in the presented simulations. In the fluid region, flameFoam solves Navier-Stokes, turbu-

lence and combustion equations, while in the solid region heat conductivity equation is

solved. The regions are coupled through standard OpenFOAM inter-region boundary

condition for temperature compressible::turbulentTemperatureCoupledBAffleMixed.

Initial conditions were set according to the experiment. Turbulence was modeled us-

ing the k-ε RANS model. Negligible values for initial turbulence parameters were se-

lected. The boundary condition for temperature on the external side of the solid grid was

set to 23 °C.

Figure 8 presents the flame propagation velocity profile (obtained from flame arrival

times) obtained by both simulations. Results are very similar, showing that the DNN

model hasn’t introduced any regressions compared to the Malet correlation around the

experimental conditions [52]. There was no significant difference of computation time be-

tween Malet and DNN calculations.

Figure 8. Vertical flame propagation velocity profiles in ENACCEF2 facility.

Figure 9 presents pressure evolutions at three different heights of the facility, time-

shifted to facilitate comparison. Pressure evolutions corresponding to the turbulent com-

bustion phase (accelerated flame) are also very similar in both cases. Differences between

evolutions obtained with Malet and DNN seem to be mostly related to a slower (quasi-

)laminar combustion phase and, consequentially, a relatively stronger impact of turbulent

acceleration, which results in a higher peak pressure wave value. At a 4 m height, the peak

value obtained with DNN is slightly lower; however, this seems to be a local result caused

by coincidental interference, since further, at 6.2 m (Figure 10), DNN results already show

higher pressure values.

Figure 8. Vertical ﬂame propagation velocity proﬁles in ENACCEF2 facility.

Figure 9presents pressure evolutions at three different heights of the facility, time-

shifted to facilitate comparison. Pressure evolutions corresponding to the turbulent com-

bustion phase (accelerated ﬂame) are also very similar in both cases. Differences between

evolutions obtained with Malet and DNN seem to be mostly related to a slower

(quasi-)

laminar combustion phase and, consequentially, a relatively stronger impact of turbulent

acceleration, which results in a higher peak pressure wave value. At a 4 m height, the peak

value obtained with DNN is slightly lower; however, this seems to be a local result caused

by coincidental interference, since further, at 6.2 m (Figure 10), DNN results already show

higher pressure values.

Figure 10 presents pressure evolutions at a height of 6.227 m. Initial shockwave

propagation and reﬂected shockwave arrival are visible. Results are unshifted in time to

show the delay in the laminar phase simulation obtained with the DNN model compared to

the Malet correlation. The DNN model provides a longer duration of quasi-laminar phase

(overall slower ﬂame propagation rate until the ﬁrst obstacle). However, ﬂameFoam with

the TFC model does not simulate a quasi-laminar regime correctly [

52

]. The assumption is

made that in turbulent cases, the main transient is controlled by the turbulent regime and

the inﬂuence of the laminar phase is limited to a larger or smaller shift in time of the onset

of the turbulent phase. This is partially illustrated in this case in Figures 9and 10 since

different methods of laminar ﬂame speed estimation led to different evolution and duration

of the quasi-laminar regime; however, when ﬂame accelerates due to turbulence, pressure

evolutions become very similar, even including the shockwave propagation and reﬂection.

Appl. Sci. 2022,12, 7460 11 of 16

Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 18

Figure 9. Pressure evolutions at different heights of ENACCEF2 (time-shifted).

Figure 10 presents pressure evolutions at a height of 6.227 m. Initial shockwave prop-

agation and reflected shockwave arrival are visible. Results are unshifted in time to show

the delay in the laminar phase simulation obtained with the DNN model compared to the

Malet correlation. The DNN model provides a longer duration of quasi-laminar phase

(overall slower flame propagation rate until the first obstacle). However, flameFoam with

the TFC model does not simulate a quasi-laminar regime correctly [52]. The assumption

is made that in turbulent cases, the main transient is controlled by the turbulent regime

and the influence of the laminar phase is limited to a larger or smaller shift in time of the onset

of the turbulent phase. This is partially illustrated in this case in Figures 9 and 10 since different

methods of laminar flame speed estimation led to different evolution and duration of the

quasi-laminar regime; however, when flame accelerates due to turbulence, pressure evolu-

tions become very similar, even including the shockwave propagation and reflection.

Figure 9. Pressure evolutions at different heights of ENACCEF2 (time-shifted).

Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 18

Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.

Figure 11 presents flow streamlines and laminar burning velocity values calculated

in both cases when the flame is in a similar location. Only values on the flame brush (pro-

gress variable source term larger than 1000 1/s) are shown. There is a clear difference in

the laminar flame velocity values obtained with both methods. The influence of pressure

and temperature are far more pronounced in the Malet correlation case, where the burnt-

mixture side of the flame brush exhibits two to three times higher velocity values than the

leading side or whole brush in the case of the DNN model. Consequently, a very thin

flame brush is obtained with Malet correlation, since combustion is completed at a higher

rate. However, at the leading side of the brush, both methods seem to produce similar

values of laminar burning velocity, corresponding to the similar overall flame propaga-

tion rate in both cases (Figure 8); higher values at the trailing side mostly only impact the

flame brush thickness.

Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.

To validate the DNN model at a somewhat higher concentration, where the Malet

correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion

in the vented laboratory-scale chamber of the University of Sydney (US) [54] was per-

formed.

The US chamber is a rectangular box of 5 × 5 × 25 cm dimensions with an open top.

Flame acceleration can be achieved with various configurations of obstacles. In the

Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.

Figure 11 presents ﬂow streamlines and laminar burning velocity values calculated

in both cases when the ﬂame is in a similar location. Only values on the ﬂame brush

(progress variable source term larger than 1000 1/s) are shown. There is a clear difference

in the laminar ﬂame velocity values obtained with both methods. The inﬂuence of pressure

and temperature are far more pronounced in the Malet correlation case, where the burnt-

mixture side of the ﬂame brush exhibits two to three times higher velocity values than the

leading side or whole brush in the case of the DNN model. Consequently, a very thin ﬂame

brush is obtained with Malet correlation, since combustion is completed at a higher rate.

However, at the leading side of the brush, both methods seem to produce similar values

of laminar burning velocity, corresponding to the similar overall ﬂame propagation rate

in both cases (Figure 8); higher values at the trailing side mostly only impact the ﬂame

brush thickness.

Appl. Sci. 2022,12, 7460 12 of 16

Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 18

Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.

Figure 11 presents flow streamlines and laminar burning velocity values calculated

in both cases when the flame is in a similar location. Only values on the flame brush (pro-

gress variable source term larger than 1000 1/s) are shown. There is a clear difference in

the laminar flame velocity values obtained with both methods. The influence of pressure

and temperature are far more pronounced in the Malet correlation case, where the burnt-

mixture side of the flame brush exhibits two to three times higher velocity values than the

leading side or whole brush in the case of the DNN model. Consequently, a very thin

flame brush is obtained with Malet correlation, since combustion is completed at a higher

rate. However, at the leading side of the brush, both methods seem to produce similar

values of laminar burning velocity, corresponding to the similar overall flame propaga-

tion rate in both cases (Figure 8); higher values at the trailing side mostly only impact the

flame brush thickness.

Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.

To validate the DNN model at a somewhat higher concentration, where the Malet

correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion

in the vented laboratory-scale chamber of the University of Sydney (US) [54] was per-

formed.

The US chamber is a rectangular box of 5 × 5 × 25 cm dimensions with an open top.

Flame acceleration can be achieved with various configurations of obstacles. In the

Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.

To validate the DNN model at a somewhat higher concentration, where the Malet

correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion in

the vented laboratory-scale chamber of the University of Sydney (US) [

54

] was performed.

The US chamber is a rectangular box of 5

×

5

×

25 cm dimensions with an open

top. Flame acceleration can be achieved with various conﬁgurations of obstacles. In the

simulated experiment, three rows of bafﬂes and a small upper obstacle were used, the

so-called conﬁguration BBBS (Figure 12a). The ﬂame is ignited at the bottom center and

propagates upwards.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 14 of 18

simulated experiment, three rows of baffles and a small upper obstacle were used, the so-

called configuration BBBS (Figure 12a). The flame is ignited at the bottom center and propa-

gates upwards.

The computational mesh (Figure 12b) of the facility was composed of the structured

orthogonal grid of the chamber-free volume and additional volume outside the top open-

ing to simulate surroundings. 2D axisymmetric mesh with 0.25 mm cell sizes was used in

the presented simulations. Around the obstacles, the mesh was graded up to 0.03125 mm

(Figure 12c).

(a) (b) (c)

Figure 12. Vented chamber and computational grid: (a) chamber scheme, (b) overall mesh view and

(c) computational mesh details below the upper obstacle.

Initial conditions were set according to the experiment—20 °C temperature and

100,000 Pa pressure. Turbulence was modeled using the k-ω-SST RANS model. Negligible

values for initial turbulence parameters were selected. The adiabatic boundary condition

for temperature was set.

Figure 13 shows overpressure evolution at the facility’s bottom. Results obtained by

flameFoam–ANN simulation are close to the experimental, including the shape of the

pressure curve. However, overpressure values are slightly over-predicted. A possible ex-

planation for the over-prediction could be missing support for quenching in the flame-

Foam, which, if implemented, could slightly decrease the average combustion rate in tur-

bulent conditions and by the obstacles and wall surfaces. Nevertheless, obtained results

indicate that the developed DNN model and its implementation in flameFoam are suita-

ble for the simulation of turbulent combustion in similar conditions.

Figure 12.

Vented chamber and computational grid: (

a

) chamber scheme, (

b

) overall mesh view and

(c) computational mesh details below the upper obstacle.

The computational mesh (Figure 12b) of the facility was composed of the structured

orthogonal grid of the chamber-free volume and additional volume outside the top opening

to simulate surroundings. 2D axisymmetric mesh with 0.25 mm cell sizes was used in

the presented simulations. Around the obstacles, the mesh was graded up to 0.03125 mm

(Figure 12c).

Initial conditions were set according to the experiment—20

◦

C temperature and

100,000 Pa

pressure. Turbulence was modeled using the k-

ω

-SST RANS model. Neg-

ligible values for initial turbulence parameters were selected. The adiabatic boundary

condition for temperature was set.

Figure 13 shows overpressure evolution at the facility’s bottom. Results obtained

by ﬂameFoam–ANN simulation are close to the experimental, including the shape of the

Appl. Sci. 2022,12, 7460 13 of 16

pressure curve. However, overpressure values are slightly over-predicted. A possible expla-

nation for the over-prediction could be missing support for quenching in the ﬂameFoam,

which, if implemented, could slightly decrease the average combustion rate in turbulent

conditions and by the obstacles and wall surfaces. Nevertheless, obtained results indicate

that the developed DNN model and its implementation in ﬂameFoam are suitable for the

simulation of turbulent combustion in similar conditions.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 18

Figure 13. Numerical and experimental [54] overpressure evolution.

Figure 14 presents flow streamlines and laminar burning velocity values after the

flame has passed the upper obstacle. Only values on the flame brush are shown. In this

case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—

the predicted values do not significantly vary across the flame brush width. Obtained LBV

values are not large and a thicker flame brush is also obtained. However, different LBV

values are obtained in different locations of the facility, though the variation is not large.

This is explained through the dependency on the pressure, which also varies in the facility

as shown by the color of the streamlines. Due to venting at the facility top, there is an

overall pressure gradient in the facility with the largest pressure at the bottom, and high-

est values reaching around 166,000 Pa. This pressure variation results in different LBV

values predicted by the DNN at different heights of the facility.

Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.

5. Conclusions

A developed DNN able to predict hydrogen LBV in dry air was developed based on

a larger DNN. The accuracy of fast DNN was tested against a testing set of data and values

from the literature. Additionally, obtained LVB curve smoothness was checked. Predic-

tions of developed DNN showed proper behavior and an R-squared value of approximately

0.997. Speed improvement of around 30 times compared to the original DNN was obtained.

Figure 13. Numerical and experimental [54] overpressure evolution.

Figure 14 presents ﬂow streamlines and laminar burning velocity values after the

ﬂame has passed the upper obstacle. Only values on the ﬂame brush are shown. In this

case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—the

predicted values do not signiﬁcantly vary across the ﬂame brush width. Obtained LBV

values are not large and a thicker ﬂame brush is also obtained. However, different LBV

values are obtained in different locations of the facility, though the variation is not large.

This is explained through the dependency on the pressure, which also varies in the facility

as shown by the color of the streamlines. Due to venting at the facility top, there is an

overall pressure gradient in the facility with the largest pressure at the bottom, and highest

values reaching around 166,000 Pa. This pressure variation results in different LBV values

predicted by the DNN at different heights of the facility.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 18

Figure 13. Numerical and experimental [54] overpressure evolution.

Figure 14 presents flow streamlines and laminar burning velocity values after the

flame has passed the upper obstacle. Only values on the flame brush are shown. In this

case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—

the predicted values do not significantly vary across the flame brush width. Obtained LBV

values are not large and a thicker flame brush is also obtained. However, different LBV

values are obtained in different locations of the facility, though the variation is not large.

This is explained through the dependency on the pressure, which also varies in the facility

as shown by the color of the streamlines. Due to venting at the facility top, there is an

overall pressure gradient in the facility with the largest pressure at the bottom, and high-

est values reaching around 166,000 Pa. This pressure variation results in different LBV

values predicted by the DNN at different heights of the facility.

Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.

5. Conclusions

A developed DNN able to predict hydrogen LBV in dry air was developed based on

a larger DNN. The accuracy of fast DNN was tested against a testing set of data and values

from the literature. Additionally, obtained LVB curve smoothness was checked. Predic-

tions of developed DNN showed proper behavior and an R-squared value of approximately

0.997. Speed improvement of around 30 times compared to the original DNN was obtained.

Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.

Appl. Sci. 2022,12, 7460 14 of 16

5. Conclusions

A developed DNN able to predict hydrogen LBV in dry air was developed based on a

larger DNN. The accuracy of fast DNN was tested against a testing set of data and values

from the literature. Additionally, obtained LVB curve smoothness was checked. Predictions

of developed DNN showed proper behavior and an R-squared value of approximately

0.997. Speed improvement of around 30 times compared to the original DNN was obtained.

Developed DNN was implemented into a combustion solver ﬂameFoam for the

prediction of LBV values. Testing CFD–DNN simulations were performed and compared

with simulations using the established Malet correlation of LBV. The comparison showed

similar results with the DNN model predicting more uniform and lower values of LBV.

Comparison against a selected experiment also showed the validity of the implemented

DNN model.

Further planned work is to perform validation of developed CFD–DNN modeling

against the experimental data.

Author Contributions:

Conceptualization, A.A. and M.P.; methodology, A.A. and M.P.; software,

A.A. and M.P.; validation, A.A. and M.P.; formal analysis, A.A. and M.P.; investigation, A.A. and M.P.;

resources, M.P.; data curation, A.A.; writing—original draft preparation, A.A.; writing—review and

editing, M.P.; visualization, A.A. and M.P.; supervision, M.P.; project administration, M.P. All authors

have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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