Fluid flows are typically described by governing equations that are too complex to be solved analytically. Consequently, numerical solutions of these equations are widely used to analyze and predict flow behavior in complex systems. The quality of numerical results depends critically on the performance of the computational/numerical methods employed. Although significant progress has been made in the development of computational methods for fluid flow, further research is still required to improve accuracy, reliability, and other related aspects. This Special Issue presents recent advances and achievements in this field.
The paper by Contribution 1 presents a comprehensive comparison of the computational efficiency and accuracy of bi-directional evolutionary structural optimization (BESO) and proportional topology optimization (PTO) for stress minimization problems. The results are problem-dependent, and the conclusions clearly outline the respective advantages and disadvantages of each method across different stress-related optimization scenarios.
The paper by Contribution 2 investigates the linear stability of bedform evolution using a one-dimensional, vertically averaged moment approach. A small-amplitude perturbation analysis is applied to an initially flat bed, ensuring that the results are relevant to conditions observed in natural systems. The limitations of the analysis, along with possible extensions, are discussed in Section 6.
The paper by Contribution 3 investigates fluid flow through an inclined channel under isothermal boundary conditions. The governing equations for velocity, temperature, and entropy generation are derived using the Casson constitutive model in conjunction with the first and second laws of thermodynamics. The effects of various fluid parameters on the flow behavior, heat transfer, and entropy production rate are examined and reported.
The paper by Contribution 4 presents a mathematical analysis of the combined effects of chemical reactions, variable fluid properties, and thermal stability on a hydromagnetic couple-stress fluid flowing through a microchannel driven by both electro-osmosis and a pressure gradient. Numerical solutions are obtained using the spectral Chebyshev collocation method and validated via the shooting Runge–Kutta method. The contributions and limitations of the study are discussed in Section 6.
The paper by Contribution 5 examines the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A mathematical model for two-phase fluid mass transfer is developed based on the classical laws of continuum mechanics and Darcy’s law, incorporating fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is obtained using the finite-difference method combined with a splitting scheme for physical processes and spatial coordinates. The accuracy of the numerical results is validated against existing real data, with errors not exceeding 8%.
Contribution 6 verifies the accuracy of a two-dimensional adaptive mesh refinement method with an arbitrary finite number of refinements using lid-driven cavity flow. The refinement criterion is derived from the qualitative theory of differential equations. The algorithm of the adaptive mesh refinement method is developed by combining this qualitative refinement criterion with existing quantitative/numerical schemes. The authors conduct extensive validation of their proposed two- and three-dimensional adaptive mesh refinement methods, evaluating accuracy, computational complexity, and performance across different numerical schemes. These methods are applicable to incompressible and steady fluid flow problems.
The Ericksen–Leslie model is widely used to study the dynamics of liquid crystal materials. Contribution 7 investigates two fully discrete finite element schemes with second-order accuracy in time for a modified Ericksen–Leslie model. The numerical solutions are obtained using a convex splitting method to handle the nonlinear potentials, combined with the second-order backward differentiation formula and the leapfrog method for temporal discretization, and the finite element method for spatial discretization. Numerical results demonstrate that both schemes achieve second-order convergence in time and space. The experiments also indicate that the leapfrog scheme is more computationally efficient than the backward differentiation formula.
The paper by Contribution 8 proposes an entropy-stable finite difference scheme for nonlinear hyperbolic systems of conservation laws. The scheme not only preserves discrete entropy inequality but also achieves high-order accuracy. The results demonstrate that it captures steep discontinuous transitions while maintaining high-order accuracy. However, its application is limited to uniform meshes or smoothly transformed curved meshes in two-dimensional cases.
Contribution 9 investigates the unconditional superconvergence error analysis of a semi-implicit Euler scheme combined with a low-order conforming mixed finite element discretization for the time-dependent Navier–Stokes equations. The numerical solution in the L∞-norm is obtained using an error-splitting technique together with high-accuracy error estimates for the element pair on rectangular meshes, without imposing any time-step restrictions. Unconditional superclose error estimates are derived through a rigorous treatment of the trilinear term. Global superconvergence results are then achieved via an interpolation post-processing approach.
As Guest Editor of this Special Issue, I sincerely thank all authors for their valuable contributions. I also extend my appreciation to the reviewers for their insightful and constructive comments, which greatly enhanced the quality of the submitted manuscripts. The objective of this Special Issue was to attract quality and novel research in the field of Advanced Computational Methods for Fluid Dynamics and Applications. We hope that the selected papers will have a meaningful impact on the international scientific community and will stimulate further research into computational methods for addressing complex problems across a wide range of disciplines and application areas.
Conflicts of Interest
The author declares no conflicts of interest.
List of Contributions
- Zheng, Y.; Qiu, J.; Chen, Z. Numerical Comparisons of Heuristic Stress Topology Optimization Methods. Mathematics 2025, 13, 2353. https://doi.org/10.3390/math13152353.
- Elgamal, M.H.; Mumtaz, M.A. Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models. Mathematics 2025, 13, 1997. https://doi.org/10.3390/math13121997.
- Njor, B.E; Lebelo, R.S; Adesanya, S.O. Irreversibility Analysis of Hydromagnetic Casson Fluid Flow Through an Inclined Channel with Isothermal Boundary Conditions. Mathematics 2025, 13, 1208. https://doi.org/10.3390/math13071208.
- Banjo, P.O.; Lebelo, R.S.; Adesanya, S.O.; Unuabonah, E.I. Energy Efficacy Enhancement in a Reactive Couple-Stress Fluid Induced by Electrokinetics and Pressure Gradient with Variable Fluid Properties. Mathematics 2025, 13, 615. https://doi.org/10.3390/math13040615.
- Uzyanbaev, R.M.; Bobremeva, Y.O.; Poveshchenko, Y.A.; Podryga, V.O.; Polyakov, S.V.; Rahimly, P.I.; Gubaydullin, I. Numerical Modeling of Two-Phase Fluid Filtration for Carbonate Reservoir in Two-Dimensional Formulation. Mathematics 2024, 12, 3412. https://doi.org/10.3390/math12213412.
- Lal, R.; Li, Z.; Li, M. Accuracy Verification of a 2D Adaptive Mesh Refinement Method by the Benchmarks of Lid-Driven Cavity Flows with an Arbitrary Number of Refinements. Mathematics 2024, 12, 2831. https://doi.org/10.3390/math12182831.
- Liao, C.; Wang, D.; Zhang, H. The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model. Mathematics 2024, 12, 672. https://doi.org/10.3390/math12050672.
- Zhang, Z.; Zhou, X.; Li, G.; Qian, S.; Niu Q. A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws. Mathematics 2023, 11, 2604. https://doi.org/10.3390/math11122604.
- Meng, X.; Yang, H. Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations. Mathematics 2023, 11, 1945. https://doi.org/10.3390/math11081945.
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