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Mathematics
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Advanced Computational Methods for Fluid Dynamics and Applications
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Advanced Computational Methods for Fluid Dynamics and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 5656

Special Issue Editor

Dr. Zhenquan Li

E-Mail Website
Guest Editor
School of Computing and Mathematics, Charles Sturt University, Thurgoona, NSW 2640, Australia
Interests: computational methods for fluid flow or computational fluid dynamics; mathematical modelling; fuzzy neural networks

Special Issue Information

Dear Colleagues,

Fluid flows are commonplace; however, we still cannot solve most fluid mechanical problems without the use of computational methods. Over the last thirty years, there have been significant progresses in computational methods for fluid flow. This Special Issue presents the recent achievements in the area. 

Any theoretical research and practical applications in computational methods for fluid flows are welcome. The topics may include mesh generation, numerical methods for the mathematical models in fluid dynamics, experimental studies showing the validity of computational methods for fluid flows, and numerical simulations of fluid flows.

Dr. Zhenquan Li
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • computational methods for fluid flow
  • numerical simulation of fluid flow
  • mathematical modelling

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Published Papers (5 papers)

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Research

22 pages, 5109 KiB  
Article
Numerical Modeling of Two-Phase Fluid Filtration for Carbonate Reservoir in Two-Dimensional Formulation
by Ravil M. Uzyanbaev, Yuliya O. Bobreneva, Yury A. Poveshchenko, Viktoriia O. Podryga, Sergey V. Polyakov, Parvin I. Rahimly and Irek M. Gubaydullin
Mathematics 2024, 12(21), 3412; https://doi.org/10.3390/math12213412 - 31 Oct 2024
Viewed by 457
Abstract
This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this [...] Read more.
This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)
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20 pages, 16548 KiB  
Article
Accuracy Verification of a 2D Adaptive Mesh Refinement Method by the Benchmarks of Lid-Driven Cavity Flows with an Arbitrary Number of Refinements
by Rajnesh Lal, Zhenquan Li and Miao Li
Mathematics 2024, 12(18), 2831; https://doi.org/10.3390/math12182831 - 12 Sep 2024
Viewed by 717
Abstract
The lid-driven cavity flow problem stands as a widely recognized benchmark in fluid dynamics, serving to validate CFD algorithms. Despite its geometric simplicity, the lid-driven cavity flow problem exhibits a complex flow regime primarily characterized by the formation of vortices at the centre [...] Read more.
The lid-driven cavity flow problem stands as a widely recognized benchmark in fluid dynamics, serving to validate CFD algorithms. Despite its geometric simplicity, the lid-driven cavity flow problem exhibits a complex flow regime primarily characterized by the formation of vortices at the centre and corners of the square domain. This study evaluates the accuracy of the 2D velocity-driven adaptive mesh refinement (2D VDAMR) method in estimating vortex centres in a steady incompressible flow within a 2D square cavity. The VDAMR algorithm allows for an arbitrary number of finite mesh refinements. Increasing the number of successive mesh refinements results in more accurate outcomes. In this paper, the initial coarse uniform grid mesh was refined ten times for Reynolds numbers 100≤Re≤7500. Results show that VDAMR accurately identifies vortex centres, with its findings closely aligning with benchmark data from six literature sources. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)
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23 pages, 1535 KiB  
Article
The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model
by Cheng Liao, Danxia Wang and Haifeng Zhang
Mathematics 2024, 12(5), 672; https://doi.org/10.3390/math12050672 - 25 Feb 2024
Viewed by 754
Abstract
In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula [...] Read more.
In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)
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18 pages, 5119 KiB  
Article
A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws
by Zhizhuang Zhang, Xiangyu Zhou, Gang Li, Shouguo Qian and Qiang Niu
Mathematics 2023, 11(12), 2604; https://doi.org/10.3390/math11122604 - 7 Jun 2023
Viewed by 1492
Abstract
The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical [...] Read more.
The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)
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17 pages, 3880 KiB  
Article
Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations
by Xiaoling Meng and Huaijun Yang
Mathematics 2023, 11(8), 1945; https://doi.org/10.3390/math11081945 - 20 Apr 2023
Cited by 1 | Viewed by 1329
Abstract
In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh [...] Read more.
In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh and the unconditional boundedness of the numerical solution in L∞-norm, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are derived firstly by dealing with the trilinear term carefully and skillfully. Then, the global superconvergence results are obtained with the aid of the interpolation post-processing technique. Finally, some numerical experiments are carried out to support the theoretical findings. Full article
(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)
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