Sparse pipeline wall information-based data-driven reconstruction for solid-liquid two-phase flow in flexible vibrating pipelines

Abstract

Deep-sea mineral resource transportation predominantly utilizes hydraulic pipeline methodology. Environmental factors induce vibrations in flexible pipelines, thereby affecting the internal flow characteristics. Therefore, real-time monitoring of solid-liquid two-phase flow in pipelines is crucial for system maintenance. This study develops an autoencoder-based deep learning framework to reconstruct three-dimensional solid-liquid two-phase flow within flexible vibrating pipelines utilizing sparse wall information from sensors. Within this framework, separate X-model and F-model with distinct hidden-layer structures are established to reconstruct the coordinates and flow field information on the computational domain grid of the pipeline under traveling wave vibration. Following hyperparameter optimization, the models achieved high reconstruction accuracy, demonstrating R² values of 0.990 and 0.945, respectively. The models’ robustness is evaluated across three aspects: vibration parameters, physical fields, and vibration modes, demonstrating good reconstruction performance. Results concerning sensors show that 20 sensors (0.06% of total grids) achieve a balance between accuracy and cost, with superior accuracy obtained when arranged along the full length of the pipe compared to a dense arrangement at the front end. The models exhibited a signal-to-noise ratio tolerance of approximately 27 dB, with reconstruction accuracy being more affected by sensor failures at both ends of the pipeline.

Keywords

Particles / Solid-liquid two-phase flow / Vibration / Flexible pipelines / Deep learning / Reconstruction

Acknowledgments

The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China (Nos. 52471293 and 12372270), the National Youth Science Foundation of China (Nos. 52101322 and 52108375), and the Program for Intergovernmental International S&T Cooperation Projects of Shanghai Municipality, China (Nos. 24510711100 and 22160710200). The Oceanic Inter-disciplinary Program of Shanghai Jiao Tong University (No. SL2022PT101) is also gratefully acknowledged. This project is also funded by the Open Fund of the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology (No. LP2415). And the National Key R&D Program of China (No. 2023YFC2811600) is gratefully acknowledged.

References

[1] Ren WL, Zhang XH, Zhang Y, Lu XB. The impact of particle size and concentration on the characteristics of solid-fluid two-phase flow in vertical pipe under pulsating flow. Chem Eng J 2024;500:156398.

[2] Wan CY, Xiao SP, Zhou D, Zhu HB, Bao Y, Kakanda K, Han Z. Numerical simulation on transport behavior of gradated coarse particles in deep-sea vertical pipe transportation. Phys Fluids 2023; 35(4):043328.

[3] Zheng MQ, Liu JC, Tian LD, Yan ZY, Zhou CJ, Li HH, Zheng CH, Chen JJ, Zheng HD. Unbaffled mesoscale reactor coupled oscillatory flow-enhanced liquid-solid two-phase flow. Powder Technol 2024;434:119292.

[4] Lareo CA, Fryer PJ. Vertical flows of solid-liquid food mixtures. J Food Eng 1998; 36(4):417-43.

[5] Pryich GP, Daugulis AJ.A novel solid-liquid two-phase partitioning bioreactor for the enhanced bioproduction of 3-methylcatechol. Biotechnol Bioeng 2007; 98(5):1008-16.

[6] Xu HL, Chen W, Hu WG. Hydraulic transport flow law of natural gas hydrate pipeline under marine dynamic environment. Eng Appl Comput Fluid Mech 2020; 14(1):507-21.

[7] Williamson CHK, Govardhan R. Vortex-induced vibrations. Annu Rev Fluid Mech 2004; 36(1):413-55.

[8] Blevins RD. Flow-induced vibration. New York: Van Nostrand Reinhold; 1977.

[9] Vandiver JK. Dimensionless parameters important to the prediction of vortex-induced vibration of long, flexible cylinders in ocean currents. J Fluids Struct 1993; 7(5):423-55.

[10] Van Atta CW, Gharib M. Ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers. J Fluid Mech 1987;174:113-33.

[11] Newman DJ, Karniadakis GE. A direct numerical simulation study of flow past a freely vibrating cable. J Fluid Mech 1997;344:95-136.

[12] Bourguet R, Karniadakis GE, Triantafyllou MS. Vortex-induced vibrations of a long flexible cylinder in shear flow. J Fluid Mech 2011;677:342-82.

[13] Zhou MM, Wang S, Kuang SB, Luo K, Fan JR, Yu AB. CFD-DEM modelling of hydraulic conveying of solid particles in a vertical pipe. Powder Technol 2019;354:893-905.

[14] Wan CY, Xiao SP, Zhou D, Zhu HB, Bao Y, Li TP, Tu JH, Kyazze MS, Han Z. Numerical analysis of coarse particle two-phase flow in deep-sea mining vertical pipe transport with forced vibration. Ocean Eng 2024;301:117550.

[15] Ren WL, Zhang XH, Zhang Y, Lu XB. The impact of pulsating parameters on particle dynamics in vertical pipe during hydraulic conveying with pulsating inlet flow. Powder Technol 2024;438:119655.

[16] Fukami K, Fukagata K, Taira K. Super-resolution analysis via machine learning: A survey for fluid flows. Theor Comput Fluid Dyn 2023; 37(4):421-44.

[17] Chen YQ, Wang D, Feng DC, Tian G, Gupta V, Cao RJ, Wan M, Chen S. Three-dimensional spatiotemporal wind field reconstruction based on Lidar and multi-scale PINN. Appl Energy 2025;377:124577.

[18] Bucchignani E. A wind field reconstruction from numerical weather prediction data based on a meteo particle model. Meteorology 2024; 3(1):70-82.

[19] Fablet R, Febvre Q, Chapron B.Multimodal 4DVarNets for the reconstruction of sea surface dynamics from SST-SSH synergies. IEEE Trans Geosci Remote Sens 2023;61:1-14.

[20] Scabini LFS, Bruno OM. Structure and performance of fully connected neural networks: Emerging complex network properties. Physica A 2023;615:128585.

[21] Li ZW, Liu F, Yang WJ, Peng SH, Zhou J. A survey of convolutional neural networks: Analysis, applications, and prospects. IEEE Trans Neural Networks Learn Syst 2021; 33(12):6999-7019.

[22] Li PZ, Pei Y, Li JQ. A comprehensive survey on design and application of autoencoder in deep learning. Appl Soft Comput 2023;138:110176.

[23] Luo ZH, Wang LY, Xu J, Chen M, Yuan JP, Tan ACC. Flow reconstruction from sparse sensors based on reduced-order autoencoder state estimation. Phys Fluids 2023; 35(7):075127.

[24] Huang QY, Zhu W, Ma F, Liu Q, Wen J, Chen L. Sparse learning model with embedded RIP conditions for turbulence super-resolution reconstruction. Comput Methods Appl Mech Eng 2024;425:116965.

[25] Xu YH, Sha YY, Wang C, Wei YJ. Adaptive estimation model: Robust full-state prediction through sparse observations with variable layout and quantity. Ocean Eng 2024;301:117617.

[26] Luo ZH, Wang LY, Xu J, Yuan JP, Chen M, Li Y, Tan ACC. A deep learning framework for reconstructing experimental missing flow field of hydrofoil. Ocean Eng 2024;293:116605.

[27] Sofos F, Sofiadis G, Chatzoglou E, Palasis A, Karakasidis TE, Liakopoulos A. From sparse to dense representations in open channel flow images with convolutional neural networks. Inventions 2024; 9(2):27.

[28] Solera-Rico A, Sanninguel Vila C, Gomez-López M, Wang Y, Almashjary A, Dawson STM, Vinuesa R. b-variational autoencoders and transformers for reduced-order modelling of fluid flows. Nat Commun 2024; 15(1):1361.

[29] Gao H, Qian WX, Dong JK, Liu J. Machine learning-based reduced-order reconstruction method for flow fields. Energ Buildings 2024;320:114575.

[30] Chen H, Guo MM, Tian Y, Le JL, Zhang H, Zhong FY. Intelligent reconstruction of the flow field in a supersonic combustor based on deep learning. Phys Fluids 2022; 34(3):035128.

[31] Dubois P, Gomez T, Planckaert L, Perret L. Machine learning for fluid flow reconstruction from limited measurements. J Comput Phys 2022;448:110733.

[32] Wei L, Guo XX, Tian XL, Zhao YK. Three-dimensional autoencoder for the flow field reconstruction of an inclined circular disk. Ocean Eng 2024;299:117284.

[33] Sanjose M, Senoner JM, Jaegle F, Cuenot B, Moreau S, Poinsot T. Fuel injection model for Euler-Euler and Euler-Lagrange large-eddy simulations of an evaporating spray inside an aeronautical combustor. Int J Multiph Flow 2011; 37(5):514-29.

[34] Yimsiri S, Soga K. DEM analysis of soil fabric effects on behaviour of sand. Geotechnique 2010; 60(6):483-95.

[35] Tsuji Y, Kawaguchi T, Tanaka T. Discrete particle simulation of two-dimensional fluidized bed. Powder Technol 1993; 77(1):79-87.

[36] Li YJ, Xu Y, Thornton C. A comparison of discrete element simulations and experiments for ‘sandpiles’ composed of spherical particles. Powder Technol 2005; 160(3):219-28.

[37] Yang SL, Luo K, Fan JR, Cen KF.Particle-scale investigation of the solid dispersion and residence properties in a 3-D spout-fluid bed. AIChE J 2014; 60 (8):2788-804.

[38] Alajbegovic A, Assad A, Bonetto F, Lahey Jr RT. Phase distribution and turbulence structure for solid/fluid upflow in a pipe. Int J Multiph Flow 1994; 20(3):453-79.

[39] Toda M, Komori N, Saito S, Maeda S. Hydraulic conveying of solids through pipe bends. J Chem Eng Jpn 1972; 5(1):4-13.

[40] Xiao SP, Wan CY, Zhu HB, Zhou D, Bao Y, Huang S, Zhang M, Han Z. Reconstruction of the solid-liquid two-phase flow field in the pipeline based on limited pipeline wall information. Phys Fluids 2025; 37(2):023314.

[41] Rumelhart DE, Hinton GE, Williams RJ. Learning representations by back-propagating errors. Nature 1986;323:533-6.

[42] Dubey SR, Singh SK, Chaudhuri BB. Activation functions in deep learning: A comprehensive survey and benchmark. Neurocomputing 2022;503:92-108.

[43] He KM, Zhang XY, Ren SQ, Sun J.Deep residual learning for image recognition. In:In:Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). p.770-8.

[44] Gunther F, Fritsch S. Neuralnet: training of neural networks. The R Journal 2010; 2(1):30-8.

[45] Kumar Y, Garg P, Moudgil MR, Singh R, Wozniak M, Shafi J, Ijaz MF. Enhancing parasitic organism detection in microscopy images through deep learning and fine-tuned optimizer. Sci Rep 2024; 14(1):5753.

[46] Agliari E, Alemanno F, Aquaro M, Fachechi A. Regularization, early-stopping and dreaming: A hopfield-like setup to address generalization and overfitting. Neural Netw 2024;177:106389.

[47] Hokamura M, Uetani H, Nakaura T, Matsuo K, Morita K, Nagayama Y, Kidoh M, Yamashita Y, Ueda M, Mukasa A, Hirai T. Exploring the impact of super-resolution deep learning on MR angiography image quality. Neuroradiology 2024; 66(2):217-26.

[48] Liu ZH, Liu K, Chen XG, Ma ZK, Lv R, Wei CY, Ma K. Deep-sea rock mechanics and mining technology: State of the art and perspectives. Int J Min Sci Technol 2023; 33(9):1083-115.

[49] Guo XS, Fan N, Zheng DF, Fu CW, Wu H, Zhang YJ, Song X, Nian T. Predicting impact forces on pipelines from deep-sea fluidized slides: A comprehensive review of key factors. Int J Min Sci Technol 2024; 34(2):211-25.

[50] Ho M, El-Borgi S, Patil D, Song GB. Inspection and monitoring systems subsea pipelines: A review paper. Struct Health Monit 2020; 19(2):606-45.