Transient Flows: Mathematical Models, Laboratory Tests, Protection Elements and Systems

1. Introduction

Transient flows represent a critical topic in the field of hydraulic engineering, one that has witnessed significant progress in recent decades. These unsteady flow conditions arise from sudden changes in pipeline operations—whether due to intentional actions such as valve closures and pump starts or stops, or unexpected events such as power failures or system malfunctions. The consequences of poorly managed transients can be severe, leading to equipment failure, infrastructure damage, or even safety hazards [1]. As a result, the analysis and control of transient flows are now recognized as fundamental components in the design and operation of fluid transport systems.

Despite the considerable advancements made, transient flow analysis continues to pose scientific and engineering challenges. The transient behavior of fluid systems involves complex interactions between hydraulic phenomena, mechanical components, and operational variables [2]. To address this complexity, researchers have increasingly relied on developments in numerical modeling, computational fluid dynamics (CFD), and data-driven approaches.

This Special Issue aims to present recent advances in understanding, predicting, and mitigating transient flows. The selected contributions cover a broad range of topics, including mathematical modeling, laboratory experiments, CFD simulations, and real-world case studies. Particular emphasis is placed on the protective systems and devices used to prevent water hammer, the novel strategies employed to achieve transient control, hydraulic transients with air entrainment or water column separation, and assessments of the risk associated with these phenomena.

By compiling these diverse and high-quality contributions, this Special Issue provides readers with a comprehensive overview of current trends and future directions in transient flow research. We hope it serves as a valuable resource for researchers, engineers, and practitioners working to improve the resilience and efficiency of hydraulic systems under transient conditions.

2. Articles

In total, eight papers were published in this Special Issue, each contributing valuable insights into different aspects of transient flow analysis and control. The selected articles reflect the diversity and depth of current research in the field, ranging from advanced modeling techniques to practical applications in real-world systems. To provide a concise overview of the contents,

Table 1. Summary of the papers published in the Special Issue “Transient Flows: Mathematical Models, Laboratory Tests, Protection Elements and Systems” for the journal Water.

Title	Authors	Keywords
Application of Newton–Raphson Method for Computing the Final Air–Water Interface Location in a Pipe Water Filling	· Dalia M. Bonilla-Correa · Óscar E. Coronado-Hernández · Vicente S. Fuertes-Miquel · Mohsen Besharat · Helena M. Ramos	· filling process · Newton–Raphson method · transient flow · pipelines · water
Smoothed Particle Hydrodynamics Simulations of Porous Medium Flow Using Ergun’s Fixed-Bed Equation	· Carlos E. Alvarado-Rodríguez · Lamberto Díaz-Damacillo · Eric Plaza · Leonardo Di G. Sigalotti	· numerical methods · REV-scale simulation · pore-scale simulation · Ergun equation · packed beds · fluidized beds
Application of Machine Learning Coupled with Stochastic Numerical Analyses for Sizing Hybrid Surge Vessels on Low-Head Pumping Mains	· Ahmed M. A. Sattar · Abedalkareem Nedal Ghazal · Mohamed Elhakeem · Amgad S. Elansary · Bahram Gharabaghi	· water hammer · transient · machine learning · stochastic numerical analysis · flat profile
Analyzing Water Leakages in Parallel Pipe Systems with Rapid Regulating Valve Maneuvers	· Vicente S. Fuertes-Miquel · Alfonso Arrieta-Pastrana · Oscar E. Coronado-Hernández	· rigid water column model · mass oscillation equation · parallel pipes · valve maneuvers · water leakages · water distribution networks
Development of Pipeline Transient Mixed Flow Model with Smoothed Particle Hydrodynamics Based on Preissmann Slot Method	· Yixin Yang · Hexiang Yan · Shixun Li · Wenke Song · Fei Li · Huanfeng Duan · Kunlun Xin · Tao Tao	· transient mixed flow · Preissmann slot model · smoothed particle · pipeline systems
MOC-Z Model of Transient Cavitating Flow in Viscoelastic Pipe	· Giuseppe Pezzinga	· transients · cavitation · viscoelasticity · Kelvin–Voigt models · method of characteristics
Two-Dimensional Transient Flow in a Confined Aquifer with a Cut-Off Curtain Due to Dewatering	· Guangcheng Li · Huiming Lin · Min Deng · Lu Wang · Jianxiao Wang · Fanshui Kong · Yushan Zhang · Qinggao Feng	· 2D transient flow · semi-analytical solution · confined aquifer · cut-off curtain · sensitivity analysis
Influence of Entrapped Air on Hydraulic Transients During Rapid Closure of a Valve Located Upstream and Downstream of an Air Pocket in Pressurised Pipes	· Oscar Pozos-Estrada	· hydraulic transients · rapid closure · hydraulic jump · air pocket · air bubbles

summarizes the titles, authors, and keywords of all contributions included in this Special Issue.

The study of thermodynamic behavior during filling processes in water pipes with trapped air is complex. Using the rigid water column (RWC) model, Bonilla-Correa et al. (Contribution 1) propose a computationally efficient method that is able to estimate critical variables such as the air pocket pressure and water column length. A Newton–Raphson-based formulation enables the direct computation of these values. The proposed method is validated against experimental data and shows strong agreement with a detailed mathematical model. A sensitivity analysis identifies the most influential parameters affecting the final system state.

Alvarado-Rodríguez et al. (Contribution 2) investigate fluid flow through a porous layer in a rectangular channel using Smoothed Particle Hydrodynamics (SPH) at both the pore scale and the Representative Elementary Volume (REV) scale, applying the Ergun equation. The study compares velocity profiles and pressure losses for different grain configurations. The results show that REV and pore-scale predictions align best at intermediate porosities (0.44–0.55). Significant discrepancies arise at higher porosities or non-uniform arrangements, highlighting the influence of grain geometry on flow behavior and the accuracy of the model.

Sattar et al. (Contribution 3) address surge protection challenges in low-head pipelines during pump failures, proposing a novel hybrid surge vessel with a dipping tube as an efficient solution. A stochastic numerical approach combined with machine learning—specifically genetic programming—is used to develop a predictive model for optimal vessel sizing. The model, validated with 2000 simulated cases, accurately estimates air and compression chamber volumes and offers a classification index for economical applications. The results outperform traditional design charts and highlight key influencing parameters such as the pipe diameter and static head.

Water leakages in water distribution networks are a topic of great interest. Fuertes-Miquel et al. (Contribution 4) introduce a novel mathematical model based on the mass oscillation equation to improve the prediction of water leakages in distribution systems, particularly under the influence of regulating valves. Unlike traditional Bernoulli-based approaches, this model captures the dynamic effects of system inertia. The method is applicable to parallel pipe systems and is validated through a case study involving two parallel pipelines. The results highlight increased leakage volumes in parallel configurations compared to single-pipe systems, offering engineers a more accurate tool for leak assessment and infrastructure planning.

Yang et al. (Contribution 5) introduce a novel one-dimensional numerical model that combines the Smoothed Particle Hydrodynamics (SPH) method with the Preissmann Slot Method (PSM) to simulate transient mixed flows in pipeline systems. The SPH-PSM model offers enhanced capabilities for capturing complex free-surface and pressurized flow interactions. After optimizing the empirical parameters, the model is validated against various flow regimes and shows strong agreement with the reference data. The results confirm the model’s effectiveness as a robust alternative for analyzing and understanding mixed-flow phenomena in urban drainage and pipeline systems.

Pezzinga (Contribution 6) presents a unitary method for solving transient cavitating flow in viscoelastic pipes, using the Method of Characteristics (MOC) combined with a Z-mirror numerical scheme (MOC-Z model). The approach accommodates both viscoelasticity and cavitation, extending the MOC-Z for Courant numbers less than 1 to resolve numerical instabilities. Four viscoelastic models (KV, GKV1, GKV2 and GKV3) are tested, and the results show that the KV model outperforms the more complex GKV models in several cases. This study provides insights into the optimal model selection for different flow conditions.

Another topic is the transient flow of groundwater in an aquifer. Li et al. (Contribution 7) present a two-dimensional analytical model of transient groundwater flow in a confined aquifer with a cut-off curtain, which is commonly used in foundation pit dewatering. The model considers both the dewatering well and the cut-off curtain in an anisotropic aquifer. Using Laplace and Fourier cosine transformations, the authors derive a semi-analytical solution for drawdown, which is validated against numerical data. A sensitivity analysis reveals that drawdown is highly influenced by the pumping rate, aquifer properties, and well structure, with specific sensitivities inside and outside the curtain.

Hydraulic transients with entrapped air are a particular case in which there are two fluids (water and air) in two phases (liquid and gas). Pozos-Estrada (Contribution 8) explores the dual impact of trapped air on fluid transients in pressurized pipelines, particularly during rapid valve closure. The study examines how air pockets and dispersed bubbles affect transient pressures, using both experiments and numerical simulations. The results show that air can mitigate pressure transients when the valve is downstream but exacerbate them when positioned upstream. This highlights the importance of considering entrapped air during pipeline design to optimize the management of transient pressure and prevent adverse effects, depending on the valve and pipeline configuration.

3. Conclusions

This Special Issue comprises a diverse set of studies that focus on various aspects of hydraulic transients, particularly those influenced by the presence of entrapped air in pipeline systems. The collection of papers emphasizes the significant role that transient phenomena, such as pressure surges, cavitation, and water hammer, play in the design and operation of fluid transport systems. Several of the articles specifically address the complexities of simulating two-phase flows, with a focus on understanding and mitigating the effects of air pockets and gas entrainment during transient events. These studies offer novel numerical models and experimental data that contribute to a more comprehensive understanding of transient dynamics, providing valuable insights into mitigating the risks associated with air-induced pressure variations.

One of the key contributions of this Special Issue is the exploration of computational fluid dynamics (CFD) models that are designed to simulate hydraulic transients during both filling and emptying processes in water pipelines. Some of the articles delve into the development and application of CFD models, offering a deeper understanding of flow behavior and the influence of pipeline geometry on transient responses. These models, validated through experimental testing, serve as powerful tools for engineers seeking to optimize the design and performance of pipeline systems under dynamic conditions.

Another important topic discussed is the characterization and modeling of air valves, which play a crucial role in managing transient pressures in pipeline systems. The research presented in this Special Issue highlights the need for accurate air valve sizing and effective strategies for controlling air pockets to prevent negative pressure surges and other potentially hazardous phenomena. The studies underscore the necessity of incorporating air valve dynamics into the overall design process to ensure the stability and safety of hydraulic systems.

Despite the advances in understanding hydraulic transients, it is clear that many aspects of this field require further investigation. The interaction between transient flow events and entrapped air, the development of more accurate numerical models, and the optimization of air management strategies remain ongoing challenges. As pipeline systems grow in complexity and scale, the need for more advanced, reliable methods for simulating and controlling transient phenomena becomes increasingly critical. Future research should continue to focus on the refinement of existing models, the enhancement of computational techniques, and the exploration of new solutions to improve the safety, efficiency, and resilience of fluid transport systems.