The application of abruptly enlarged flows to adjust the drag of aerodynamic vehicles using machine learning models has not been investigated previously. The process variables (Mach number (M), nozzle pressure ratio (η), area ratio (α), and length to diameter ratio (γ)) were numerically explored to address several aspects of this process, namely base pressure (

Fluid flows with sudden axisymmetric expansions are a challenging topic in fluid dynamics that may be encountered in a wide range of areas and industrial applications. In most situations, a round tube with a smooth inner surface is adopted. A drop in the pressure in the wake zone is seen while the duct area ascends rapidly. Such type of expanded flows undergoes flow separation and reattachment. Although significant work has been done, these flows are still not completely realized. The area of concern is at the nozzle-duct interface where the nature of the fluid flow phenomena is quite complicated, as it involves shock waves, expansion waves, and high pressure gradients [

Studies have been carried out by many researchers using a sudden expansion duct with ribs and splitter plates [

Following the investigative and numerical techniques to passive flow regulation, a numerical approach was also implemented to solve the suddenly expanded flow process. The computational fluid dynamic (CFD) approach is most commonly used for this type of analysis. This approach was undoubtedly used by multiple researchers associated with the current investigation. Over the previous two decades, both passive and active control approaches have been successfully used in CFD investigations. Turbulence modeling is an important aspect of fluid analysis and in most cases; a density-based model was proven to be more suitable for compressible flows. CFD analysis revealed that flow control via tiny jets was favorable for regulating pressure in the separated zone at large

Several Taguchi designs, response surface methods (RSM), and soft computing approaches have been implemented to determine the level of accuracy and reliability. A study regarding wind tunnel adjustments with variable throat diameters was conducted by Cameron et al. [_{27} orthogonal array were used to construct the dataset for training, testing, and verification. _{9} orthogonal array, and linear regression equations for

In a suddenly expanded flow process, a sudden change in the cross-sectional area of the flow from the nozzle to the expanded duct, creates massive changes in velocity and pressure of flow. The dynamics of the flow are dependent on a number of factors that include,

Based on the literature cited above, the authors found numerous works that apply different neural network computational techniques on suddenly expanded flows. Also, techniques such as, genetic algorithm and particle swarm optimization in different forms have been applied to similar fluid flow problems. The regression co-efficients

The flow process to be explored experimentally is shown in _{o}) required to develop

S. I. No. | Control parameters | Levels of the parameters | ||
---|---|---|---|---|

Low (−) | Medium (0) | High (+) | ||

1. | 2.0 | 2.5 | 3.0 | |

2. | 3 | 7 | 11 | |

3. | 3.25 | 4.75 | 6.25 | |

4. | 3 | 6 | 9 |

Nozzles having exit Mach values of 2.0, 2.5, and 3.0 were constructed for the proposed study. The nozzles’ exit Mach numbers were optimized utilizing isentropic relations by Genick [

J. E. Elman suggested introducing a context layer to a feedforward NN to create a one-step postponement manipulator for transitory memory functions, for the NN to respond to time-varying features, thereby improving network stability. The network might then be utilized to tackle quick optimization-seeking challenges that effectively represent the properties of compelling process systems, giving rise to Elman NN [

Jiang et al. [

The beetle is modeled as a parameter that receives input in an n-dimensional space and to compute values, it chooses locations close to oneself based on multiple values obtained throughout either side of the multi spatial domain to obtain the optimal global value. The simplified model was shown in Wang et al. [

Two antennae are located on either ends of the simplified center of the beetle head.

The ratio of the movement of a specific beetle

The positioning of the beetle’s head upon approaching the next place from the existing position is random.

The following are the stages of the BAS algorithm:

1. Create and standardize a T-dimensional random vector of the beetle’s original direction (Wei et al. [

2. At the ^{th} iteration, beetle takes the coordinates as stated by (Lin et al. [

^{th} iteration, respectively.

3. The intensity of the food odor is determined by the estimated value of the orientation of the beetle antennas; the next travel direction is then modified (Yue et al. [

Here, ^{th} iteration, ^{th} sample.

4. As we use the step factor

The BAS-ENN algorithm is eventually established based on the theories presented above, as illustrated in

The trained model forecasts for the fresh test set. The calculations are performed for this specified test set, and the findings obtained are output.

According to the literature, the parameters influencing the suddenly expanded flow process are numerous and complicated. However, in the flow process,

S.I. No. | Operating conditions | Output responses | ||||
---|---|---|---|---|---|---|

1. | 2.0 | 3 | 3.25 | 3 | 0.75 | 0.72 |

2. | 2.0 | 7 | 4.75 | 3 | 0.82 | 0.78 |

3. | 2.0 | 11 | 6.25 | 3 | 0.78 | 0.78 |

4. | 2.5 | 11 | 3.25 | 3 | 0.91 | 0.85 |

5. | 2.5 | 7 | 4.75 | 3 | 0.63 | 0.66 |

6. | 2.5 | 3 | 6.25 | 3 | 0.71 | 0.66 |

7. | 3.0 | 11 | 3.25 | 3 | 0.52 | 0.49 |

8. | 3.0 | 3 | 4.75 | 3 | 0.45 | 0.44 |

9. | 3.0 | 7 | 6.25 | 3 | 0.38 | 0.35 |

10. | 2.0 | 3 | 3.25 | 6 | 0.66 | 0.59 |

11. | 2.0 | 7 | 4.75 | 6 | 0.71 | 0.73 |

12. | 2.0 | 11 | 6.25 | 6 | 0.44 | 0.42 |

13. | 2.5 | 11 | 3.25 | 6 | 0.49 | 0.46 |

14. | 2.5 | 7 | 4.75 | 6 | 0.32 | 0.30 |

15. | 2.5 | 3 | 6.25 | 6 | 0.58 | 0.55 |

16. | 3.0 | 11 | 3.25 | 6 | 0.69 | 0.62 |

17. | 3.0 | 3 | 4.75 | 6 | 0.75 | 0.70 |

18. | 3.0 | 7 | 6.25 | 6 | 0.91 | 0.89 |

19. | 2.0 | 3 | 3.25 | 9 | 0.61 | 0.54 |

20. | 2.0 | 7 | 4.75 | 9 | 0.35 | 0.33 |

21. | 2.0 | 11 | 6.25 | 9 | 0.40 | 0.37 |

22. | 2.5 | 11 | 3.25 | 9 | 0.55 | 0.52 |

23. | 2.5 | 7 | 4.75 | 9 | 0.62 | 0.58 |

24. | 2.5 | 3 | 6.25 | 9 | 0.93 | 0.86 |

25. | 3.0 | 11 | 3.25 | 9 | 0.84 | 0.80 |

26. | 3.0 | 3 | 4.75 | 9 | 0.50 | 0.47 |

27. | 3.0 | 7 | 6.25 | 9 | 0.35 | 0.32 |

The actual

The confidence intervals [

Before the NN analysis, we must understand the behavior of base pressure for the different parameters. For this purpose, variation of

It is important to note that, a higher value for

The model contains multiple input parameters, as indicated in

To evaluate the performance of various algorithms on the

Type of model | Values of specific parameters |
---|---|

GA-BP | ^{−3}; ^{−2}; ^{−2}; ^{−2}; |

PSO-BP | ^{−3}; ^{−2}; |

PCA-BAS-ENN | ^{−3}; ^{−2}; |

Meanwhile, three performances of

Type of errors | Models | Training data | Testing data | ||
---|---|---|---|---|---|

Model 1 | 3.107 | 3.513 | 4.210 | 4.310 | |

Model 2 | 3.102 | 3.870 | 3.824 | 4.050 | |

Model 3 | 2.787 | 3.115 | 3.007 | 3.355 | |

Model 1 | 1.656 | 4.255 | 2.257 | 5.334 | |

Model 2 | 1.355 | 4.410 | 2.120 | 4.995 | |

Model 3 | 1.267 | 3.850 | 2.088 | 4.550 | |

Model 1 | 4.008 | 4.556 | 5.552 | 5.595 | |

Model 2 | 4.123 | 4.865 | 5.345 | 5.151 | |

Model 3 | 3.995 | 4.445 | 4.395 | 4.858 |

Note: Model 1 (GA-BP); Model 2 (PSO-BP); Model 3 (PCA-BAS-ENN).

From

The primary reason for the poor performance of the GA-BP model (

The hybrid PCA-BAS-ENN model as seen in

The frequency distribution of the absolute error has been presented in

The theoretical

Here, ^{nd} and 5^{th} order Mach numbers, respectively; ^{nd} and 5^{th} order Mach values, respectively; ^{nd} and 5^{th} order measured ^{nd} and 5^{th} order ^{nd} and 5^{th} order base pressure was 0.341 and 0.393, respectively. This means that the average

Supersonic expanded flow process has been found to be very handy in regulating the base drag of aerodynamic vehicles. Numerous experimental and numerical investigations carried out by researchers previously, supported the implementation of internal modifications in the abruptly expanded duct in order to achieve favourable performance characteristics. However, analyzing such flows for a number of input parameters becomes time-consuming, challenging, and expensive. In this regard, the use of machine learning models in non-linear fluid flow problems still remains unexplored. Therefore, the current study developed a data-driven forecasting model by employing the PCA and BAS-ENN algorithms to determine the optimal setting of

The induction of a cavity into the expanded duct decreased

Under identical settings, the PCA-BAS-ENN model was compared to the other two algorithms, for determining

Upon having integrated with the PCA-BAS-ENN model, the practical field implementation displays that the average ^{nd} and 5^{th} order

All the machine learning models of the present study for

Convergence precision

Measured value of base pressure (Pa)

Box-Behnken Design

Central composite design

Computation fluid dynamics

Input layer

Hidden layer

Context layer

Spacing between two beetle antennae (mm)

Number of iterations

Output layer

Mach number

Number of samples

Specific beetle movement

Genetic algorithm back propagation

Learning rate

Mean absolute error

Mean absolute percent error

Nueral network

Crossover probability

Nutation probability

Principal component analysis-beetle search algorithm—elman neural network

Particle swarm optimization-back propagation

Regression coefficient

Root mean square error

Population size

Highest time for training

Three error band

Location of the simplified center

Output representation of the Elman neural network

Area ratio (ratio of duct area to nozzle exit area)

Non-dimensionnal base pressure or base pressure

Step decay co-eeficient

Nozzle pressure ratio

Weights

Length to diameter ratio of the duct

Second order

Fifth order

Cavity/cavities in the expanded duct

Predicted

Actual

Desired

The authors acknowledge the support of Prince Sultan University for paying the article processing charges (APC) of this publication.

This research is supported by the Structures and Materials (S&M) Research Lab of Prince Sultan University.

The authors declare that they have no conflicts of interest to report regarding the present study.