Computational Fluid Dynamics Problems Research Articles

Comparative Analysis of NURBS and Finite Element Method in Computational Fluid Dynamics Applications: Case Study on NACA 2412 Airfoil Aerodynamics

In this research, an attempt was made to employ the Non-Uniform Rational B-Splines (NURBS) method for a challenging computational fluid dynamics (CFD) problem of aerodynamics around NACA 2412 airfoils. The comparison was carried out thoroughly by using the same boundary conditions and geometry, comparing NURBS to standard FEM implementations. Our study was interested in demonstrating the foreseeable functionalities of NURBS for solving complex CFD problems and conducting a comparative effectiveness performance evaluation between them with traditional FEM methodologies.

MH-DCNet: An improved flow field prediction framework coupling neural network with physics solver

With the development of intelligent computing technology, deep learning methods have provided an efficient solution for rapid flow field prediction in computational fluid dynamics (CFD) problems. However, existing methods have limitations in handling interference among physical variables due to different data distributions, leading to a decline in prediction performance. In this paper, we propose MH-DCNet, an improved flow field prediction framework that couples a neural network with a physics solver. Specifically, to address the data distribution problem, we design a multi-head deep convolutional neural network that decouples the prediction of physical variables through multiple encoders and decoders. We also develop a hybrid loss function by introducing the mean structural similarity to better capture the complex spatial structures and distribution features of flow fields. We evaluate MH-DCNet with unseen geometries and various flow conditions. Experimental results show that MH-DCNet outperforms other advanced models in efficiency and generalization capability. It accelerates the prediction process by 2.35 times compared to the CFD method while meeting the convergence constraints.

Application of the iteratively regularized Gauss–Newton method to parameter identification problems in Computational Fluid Dynamics

Field Inversion and Machine Learning is an active field of research in Computational Fluid Dynamics (CFD). This approach can be leveraged to obtain a closed-form correction for a given turbulence model to improve the predictions. The fundamental approach is to insert a parameter into the system of RANS equations and determine it in a way such that, for example, a given pressure distribution is better approximated compared to the one obtained with the original set of equations. The goal of this article is twofold. Numerical arguments are presented that these kinds of problems can be severely ill-posed. In the second part, an approach is presented to directly reconstruct the turbulent viscosity field along with an example. The Iteratively Regularized Gauss-Newton Method (IRGNM) is used for a realization. The construction of a problem-adapted norm for a finite volume method is presented. Finally, an outlook is presented on how this approach can be used to possibly modify or improve turbulence models such that not only one, but a larger number of test cases are considered.

Reducing spatial discretization error on coarse CFD simulations using an openFOAM-embedded deep learning framework

AbstractWe propose a method for reducing the spatial discretization error of coarse computational fluid dynamics (CFD) problems by enhancing the quality of low-resolution simulations using deep learning. We feed the model with fine-grid data after projecting it to the coarse-grid discretization. We substitute the default differencing scheme for the convection term by a feed-forward neural network that interpolates velocities from cell centers to face values to produce velocities that approximate the down-sampled fine-grid data well. The deep learning framework incorporates the open-source CFD code OpenFOAM, resulting in an end-to-end differentiable model. We automatically differentiate the CFD physics using a discrete adjoint code version. We present a fast communication method between TensorFlow (Python) and OpenFOAM (c++) that accelerates the training process. We applied the model to the flow past a square cylinder problem, reducing the error from 120% to 25% in the velocity for simulations inside the training distribution compared to the traditional solver using an x8 coarser mesh. For simulations outside the training distribution, the error reduction in the velocities was about 50%. The training is affordable in terms of time and data samples since the architecture exploits the local features of the physics.

Sample-efficient and surrogate-based design optimization of underwater vehicle hulls

Physics simulations like computational fluid dynamics (CFD) are a computational bottleneck in computer-aided design (CAD) optimization processes. To overcome this bottleneck, one requires either an optimization framework that is highly sample-efficient, or a fast data-driven proxy (surrogate model) for long-running simulations. Both approaches have benefits and limitations. Bayesian optimization is often used for sample efficiency, but it solves one specific problem and struggles with transferability; alternatively, surrogate models can offer fast and often more generalizable solutions for CFD problems, but gathering data for and training such models can be computationally demanding. In this work, we leverage recent advances in optimization and artificial intelligence (AI) to explore both of these potential approaches, in the context of designing an optimal unmanned underwater vehicle (UUV) hull. For first approach, we investigate and compare the sample efficiency and convergence behavior of different optimization techniques with a standard CFD solver in the optimization loop. For the second approach, we develop a deep neural network (DNN) based surrogate model to approximate drag forces that would otherwise be computed via the CFD solver. The surrogate model is in turn used in the optimization loop of the hull design. Our study finds that the Bayesian Optimization—Lower Condition Bound (BO-LCB) algorithm is the most sample-efficient optimization framework and has the best convergence behavior of those considered. Subsequently, we show that our DNN-based surrogate model predicts drag force on test data in tight agreement with CFD simulations, with a mean absolute percentage error (MAPE) of 1.85%. Combining these results, we demonstrate a two-orders-of-magnitude speedup (with comparable accuracy) for the design optimization process when the surrogate model is used. To our knowledge, this is the first study applying Bayesian optimization and DNN-based surrogate modeling to the problem of UUV design optimization, and we share our developments as open-source software.

Non-Fourier computations of heat and mass transport in nanoscale solid-fluid interactions using the Galerkin finite element method

PurposeTo improve the thermal performance of base fluid, nanoparticles of three types are dispersed in the base fluid. A novel theory of non-Fourier heat transfer is used for design and development of models. The thermal performance of sample fluids is compared to determine which types of combination of nanoparticles are the best for an optimized enhancement in thermal performance of fluids. This article aims to: (i) investigate the impact of nanoparticles on thermal performance; and (ii) implement the Galerkin finite element method (GFEM) to thermal problems.Design/methodology/approachThe mathematical models are developed using novel non-Fourier heat flux theory, conservation laws of computational fluid dynamics (CFD) and no-slip thermal boundary conditions. The models are approximated using thermal boundary layer approximations, and transformed models are solved numerically using GFEM. A grid-sensitivity test is performed. The accuracy, correction and stability of solutions is ensured. The numerical method adopted for the calculations is validated with published data. Quantities of engineering interest, i.e. wall shear stress, wall mass flow rate and wall heat flux, are calculated and examined versus emerging rheological parameters and thermal relaxation time.FindingsThe thermal relaxation time measures the ability of a fluid to restore its original thermal state, called thermal equilibrium and therefore, simulations have shown that the thermal relaxation time associated with a mono nanofluid has the most substantial effect on the temperature of fluid, whereas a ternary nanofluid has the smallest thermal relaxation time. A ternary nanofluid has a wider thermal boundary thickness in comparison with base and di- and mono nanofluids. The wall heat flux (in the case of the ternary nanofluids) has the most significant value compared with the wall shear stresses for the mono and hybrid nanofluids. The wall heat and mass fluxes have the highest values for the case of non-Fourier heat and mass diffusion compared to the case of Fourier heat and mass transfer.Originality/valueAn extensive literature review reveals that no study has considered thermal and concentration memory effects on transport mechanisms in fluids of cross-rheological liquid using novel theory of heat and mass [presented by Cattaneo (Cattaneo, 1958) and Christov (Christov, 2009)] so far. Moreover, the finite element method for coupled and nonlinear CFD problems has not been implemented so far. To the best of the authors’ knowledge for the first time, the dynamics of wall heat flow rate and mass flow rate under simultaneous effects of thermal and solute relaxation times, Ohmic dissipation and first-order chemical reactions are studied.

Molecular dynamics simulation of argon isochoric transition to supercritical state

AbstractThe effect of the initial atoms distribution on the molecular dynamics (MD) simulation of a model atomic fluid (argon) is investigated for the case of the isochoric phase transition to the supercritical state. In particular, the case of uniformly distributed atoms in the simulation domain is compared with the case of separated liquid and vapor atoms. The sensitivity of simulations to asymmetric nanoscale perturbations in the boundary is also studied. Despite its high computational cost, the MD approach has the potential to successfully address long‐standing problems in computational fluid dynamics (CFD), especially those associated with mathematical singularities, such as contact angles, vortices, phase transitions and so forth. Unlike conventional CFD simulations, where the initial condition is the pressure or velocity distribution in the simulation domain, MD simulations also require the initial position of each molecule. Thus, it is important to understand whether a judicious choice of the initial distribution of molecules can reduce the overall computation time of the simulation. The evolution of the model fluid system during the phase transition was simulated using a Lennard‐Jones interatomic potential, corrected with the Lorentz–Berthelot mixing rule for the interactions with the solid walls. The system was allowed to relax until equilibrium, and then a Heaviside temperature step was applied to the wall to bring the system to supercritical conditions. Results show the initial choice of the atoms distribution can significantly affect the computational time, while the effect of asymmetric perturbations on the boundary is negligible.

SYNTHETIC DATA GENERATION FOR ANN MODELING OF THE HYDRODYNAMIC PROCESSES OF IN-SITU LEACHING

The work presents an approach to enhance the forecasting capabilities of In-Situ Leaching processes during both the production stage and early prognosis. ISL, a crucial method for resource extraction, demands rapid on-site forecasting to guide the deployment of new technological blocks. Traditional modeling techniques, though effective, are hindered by their computational demands and network throughput requirements, particularly when dealing with substantial datasets or remote computing needs. The integration of AI technologies, specifically neural networks, offers a promising opportunity for expedited calculations by leveraging the power of forward propagation through pretrained neural models. However, a critical challenge lies in transforming conventional numerical datasets into a format suitable for neural modeling. Furthermore, the scarcity of training data during the production phase, where vital parameters are concealed underground, poses an additional challenge in training AI models for In-Situ Leaching processes. This research addresses these challenges by proposing a methodology for generating training data tailored to the most resource-intensive Computational Fluid Dynamics problems encountered during modeling. Traditional numerical modeling techniques are harnessed to construct training datasets comprising input and corresponding expected output data, with a particular focus on varying well network patterns. Subsequent efforts are directed at the conversion of the acquired data into a format compatible with neural networks. The data is normalized to align with the data ranges stipulated by the activation functions employed within the neural network architecture. This preprocessing step ensures that the neural model can effectively learn from the generated data, facilitating accurate forecasting of In-Situ Leaching processes. An advantage of proposed technique lies in provision of large, reliable datasets to train neural network to predict hydrodynamic properties based on technological regimes currently active or expected on ISL site. A major implication of this approach lies in applicability of pre-trained AI technologies to forecast future or determine current hydrodynamic regime in the stratum circumventing cost deterministic simulations currently deployed at mining sites. Hence, innovative approach outlined in this paper holds promise for optimizing forecasting, allowing for quicker and more efficient decision-making in resource extraction operations while getting around the computational barriers associated with traditional methods.

A hybrid CPU‐GPU paradigm to accelerate reactive CFD simulations

AbstractThe solution of reactive computational fluid dynamics (CFD) simulations is accelerated by the implementation of a hybrid central processing unit/graphics processing units (CPU/GPU) Finite Volume solver based on the operator‐splitting strategy, where the chemistry integration is treated independently of the flow solution. The integration of ordinary differential equations (ODEs) describing the finite‐rate chemical kinetics is solved by an adaptive multi‐block explicit solver on GPUs, while the load of the fluid solution is distributed on a multicore CPU algorithm. The resulting speed‐up for reactive CFD simulations is up to 10; the performance gain increases with the size of the mechanism. The proposed implementation is general and can be applied to any CFD problem where the governing equations for the fluid transport are coupled with an ODE system. Code validation is performed against reference solutions on a selection of test cases involving reacting flows.

Flow2Mesh: A flow-guided data-driven mesh adaptation framework

Mesh adaptation is crucial in numerical simulation, providing optimal resource allocation for accurately capturing physical phenomena. However, when applied to Computational Fluid Dynamics (CFD) problems with complex multi-scale properties, existing adaptation methods face huge challenges due to the high computational cost of solving auxiliary partial differential equations (PDEs) and the difficulty in aligning the flow features with mesh geometric features. In this work, an end-to-end data-driven mesh adaptation framework, Flow2Mesh, is proposed to address these challenges by adopting a hybrid modeling strategy to construct the mapping from pixelated flow-fields to graph-based meshes. It achieves a rapid and accurate one-step mesh adaptation via a perceptual feature network (PFN) and a mesh movement network (MMN). PFN extracts the global perceptual features from flow-fields to enhance flow feature representation and mesh resolution independence. In MMN, these features are utilized to deform the initial mesh to a topology-invariant adaptive mesh by a proposed physically driven mesh convolutional network. It considers the inherent mesh geometric information for efficient node feature aggregation and alignment of mesh density with a flow-field structure. To generate high-quality adaptive meshes, various mesh-related losses are designed to regularize the mesh movement and alleviate the mesh tangling. Experiments in CFD scenarios demonstrate the generalization of our model to different design parameters and mesh configurations. It takes three orders of magnitude less time to generate similar meshes than the PDE-based method. The results exhibit the potential of Flow2Mesh to be a flexible and reliable tool for rapid mesh adaptation in scientific and industrial fields.

Vectorized Numerical Algorithms to Solve Internal Problems of Computational Fluid Dynamics

The opportunities provided by new information technologies, object-oriented programming tools, and modern operating systems for solving boundary value problems in CFD described by partial differential equations are discussed. An approach to organizing vectorized calculations and implementing finite-difference methods for solving boundary value problems in CFD is considered. Vectorization in CFD problems, eliminating nested loops, is ensured through the appropriate data organization and the use of vectorized operations with arrays. The implementation of numerical algorithms with vectorized mesh structures, including access to internal and boundary mesh cells, is discussed. Specific examples are reported and the implementation of the developed computational algorithms is discussed. Despite the fact that the capabilities of the developed algorithms are illustrated by solving benchmark CFD problems, they enable a relatively simple generalization to more complex problems described by three-dimensional equations.

A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

Robust shape optimization using artificial neural networks based surrogate modeling for an aircraft wing

Aerodynamic shape optimization is a very active area of research that faces the challenges of highly demanding Computational Fluid Dynamics (CFD) problems, optimization with Partial Differential Equations (PDEs) as constraints, and the appropriate treatment of uncertainties. This includes the development of robust design methodologies that are computationally efficient while maintaining the desired level of accuracy in the optimization process. This paper addresses aerodynamic shape optimization problems involving uncertain operating conditions. After a review of possible approaches to account for uncertainties, an Artificial Neural Network (ANN) model is used to approximate the aerodynamic coefficients when the operating conditions vary. Robust optimization problem-solving approaches based on deterministic measurements are used, inspired by the work of Deb [Deb K., Gupta H. Introducing robustness in multi-objective optimization. KanGAL Report 2004–2016, Kanpur Genetic Algorithms Laboratory, Indian Institute of Technology, Kanpur, India (2004)]. The first procedure is a direct extension of a technique used for single-objective optimization. The second is a more practical approach allowing a user to define the desired degree of robustness in a problem. These approaches have been tested and validated in the case of the optimization of an aircraft wing profile in the transonic regime considering two uncertain variables: the Mach number and the angle of incidence.

On Machine-Learning-Aided Two-Scale Solution for Turbulent Fluid Flows

Scale-resolving solutions for computational fluid dynamics problems have usually been challenging due to their request for computing resources. A two-scale framework was proposed for more efficient solutions to couple a local fine-mesh solution with a global coarse-mesh solution. The methodology was successfully implemented and demonstrated for a canonical turbulent channel flow and for a tripped turbulent boundary layer. The solution mapping from the local fine-mesh to the global coarse-mesh region is realized by modifying the flow-governing equations in the under-resolved coarse-mesh region through adding extra forcing source terms generated from the space–time-averaged fine-mesh solutions. However, the high-gradient transitional region presents additional challenges when applying the Chebyshev spectral method for mapping the source terms; thus the high-gradient frontal region has not been fully resolved in the streamwise direction. In the present work, the propagation of the source terms is facilitated by machine learning tools (multilayer perceptron-based neural network) so as to implement the method in flowfields with high gradients or drastic changes in the mean velocity. The neural-network-based propagation model is shown to be capable of accurately estimating the source terms in the near-wall coarse-mesh region. The mean flow there thus can be nicely reproduced by the source-term propagation. The machine-learning tools thus provide potential as the more advanced source-term propagation method for the two-scale framework to be implemented in more complicated flowfields.

Novel design of nonlinear autoregressive with exogenous noise-based intelligent networks for dynamics of MHD Casson–Williamson models involving heat generation and viscous dissipation

In the current arena, the role of artificial intelligence (AI) has gained immense impact in the field of computational fluid dynamics (CFD) due to mesh improvement efficiency, optimal manual intervention, reliable forecasting, comprehending data analytic decisions, and utilization of machine automation. The study under investigation treatise on CFD problem representing the flow of Casson and Williamson fluids (CWFs) model with heat generation and viscous dissipation by using AI-based knacks through nonlinear autoregressive exogenous (NARX) networks backpropagated with Bayesian regularization technique (BRT), i.e. NARX-BRT. The design NARX-BRT procedure implemented on data generated with Adams numerical scheme for CWFs by carefully altering parameters i.e. Darcy number, Casson fluid, mass convective Williamson, mixed convection, thermal conductivity, thermal convective, and Eckert number, with fixed values Prandtl and Schmidt numbers. Results invariably matched with the reference/standard numerical solutions for the CWF model having significantly reduced magnitude of error for each anticipated CWF scenario. The effectiveness of projected CWF is portrayed exhaustively through mean square error (MSE) based iterative learning curves, analysis of adaptive controlling factors, error histogram plots, and regression metrics for variants of nonlinear differential order systems of CWF-based CFD.

Numerical calculation of the turbulent flow past a surface mounted cube with assimilation of PIV data

The application of Data Assimilation (DA) methods in Computational Fluid Dynamics (CFD) problems is a concept actively being explored to couple CFD with Experimental Fluid Dynamics data. Here, Particle Image Velocimetry (PIV) data are assimilated in an OpenFOAM based CFD solver to calculate the velocity and pressure fields of the turbulent flow past a surface mounted cube inside an atmospheric boundary layer for three planes belonging to the symmetry plane of the flow. At first, the SIMPLE algorithm is used to correct both pressure and velocity fields, with the PIV data used to formulate the initial and boundary conditions. The Reynolds stresses are calculated directly from the PIV data instead of using a turbulence model. Next, we use two implementations of the nudging method and two formulations of the Kalman Filter in order to assimilate the PIV data into the iterative SIMPLE procedure. A grid independence study is performed, and the performance of the different methods is assessed. The CFD predicted pressure field is in good agreement with pressure measurements on the cube surface. The results also show that the SIMPLE based correction step already leads to a significant reduction of both the mean and the variance of the continuity errors as well as the difference between the original PIV data and the resulting velocity fields. The application of the DA methods, particularly the KF, leads to minor further improvement of the results but does improve convergence of the CFD solver.

Numerical simulation and validation of laminar flow through a 2D pipe using computational fluid dynamics

In modern times, there has been a significant increase in the use of software to simulate fluid dynamics. There exist a variety of closed-source software packages that are utilized for both modeling and simulating fluid behavior. These proprietary programs have become increasingly common and are widely utilized in numerous industries that require advanced simulations. This work focuses on the use of a commercial software package designed for computational fluid dynamics (CFD) problems to model laminar flow through a pipe, with the objective of gaining a deeper understanding of the dynamic structures and interactions that occur within the laminar flow. In this paper, an accurate computation of two-dimensional flow is performed using the Time Volume Element Methodology. A grid sensitivity analysis with Richardson extrapolation is conducted to determine the grid-independent solutions, ensuring that the results obtained are reliable and accurate. The findings of this study suggest that the relationship between various parameters, such as velocity and pressure, is heavily influenced by the position and shape of the pipe through which the flow occurs. The investigation utilized a grid sensitivity analysis with Richardson extrapolation to determine the percentage error between the analytical and grid-independent solutions for different parameters, revealing that the errors are on the order of 10-1 %.

Detailed investigation on thermal enhancement and mass transport in 3D flow of Carreau–Yasuda ternary and hybrid nanofluids using the finite element method

PurposeThe purpose of this study is to analyze the transportation of heat and mass in three-dimensional (3D) shear rate-dependent viscous fluid. Thermal enhancement plays a significant role in industrial and engineering applications. For this, the authors dispersed trihybrid nanoparticles into the fluid to enhance the working fluid’s thermal enhancement.Design/methodology/approachThe finite element method is a numerical scheme and is powerful in achieving convergent and grid-independent solutions compared with other numerical techniques. This method was initially assigned to structural problems. However, it is equally successful for computational fluid dynamics problems.FindingsWall shear stress has shown an increasing behavior as the intensity of the magnetic field is increased. Simulations have predicted that Ohmic heat in the case of trihybrid nanofluid (MoS2–Al2O3–Cu/C2H6O2) has the greatest value in comparison with mono and hybrid nanofluids. The most significant influence of chemical reaction on the concentration in tri-nanofluid is noted. This observation is pointed out for both types of chemical reaction (destructive or generative) parameters.Originality/valueThrough a literature survey, the authors analyzed that no one has yet to work on a 3D magnetohydrodynamics Carreau–Yasuda trihybrid nanofluid over a stretched sheet for improving heat and mass transfer over hybrid nanofluids. Herein, molybdenum disulfide (MoS2), aluminum oxide (Al2O3) and copper (Cu) nanoparticles are mixed in ethylene glycol (C2H6O2) to study the thermal enhancement and mass transport of their corresponding resultant mono (Cu/C2H6O2), hybrid (Al2O3–Cu/C2H6O2) and trihybrid (MoS2–Al2O3–Cu/C2H6O2) nanofluids.

A Computational Fluid Dynamics Enhanced Oil Recovery Formulation Based on the Volume Average Procedure on the Full Momentum Equation: Effects of Polymer Solution Injection in Real Reservoirs with Anisotropic Permeability

Summary In this work, we propose a new methodology to simulate the process of enhanced oil recovery (EOR) in a 3D domain, considering a non-Newtonian fluid phase. The mass balance for the two phases and the balance of momentum are based on the volume averaging theory, which upscales the information on the microscale to make viable the solution in a real case. The full balance of momentum is used in place of the usually adopted Darcy’s law, and the equations are written for each phase in terms of porosity and fluid saturation. The equations are introduced in a new numerical solver developed for the OpenFOAM toolbox, which is an open-source C++ library created to simulate problems of computational fluid dynamics (CFD). First, the methodology is verified by comparing the results obtained from the volume averaging theory equations with Darcy’s law. Then, Newtonian and non-Newtonian cases for Buckley-Leverett, 2D, and 3D meshes are presented. We also present cases with a 3D mesh in a domain extracted from a real reservoir and properties of real injector fluids. The present approach is able to accommodate anisotropic permeability, heterogeneity, and non-Newtonian effects. We compare fluid saturation over time for the different cases as well as the accumulated volume of oil over time and the flow output of the domain. The simulations performed were able to demonstrate the effectiveness of polymeric solutions, comparing polyacrylamide (PAA) and polymeric surfactant (PS) cases with different concentrations, for better use in the production of available resources in reservoirs. In the five-spot cases with polymeric solutions, lower flow rates are achieved when the concentration is increased and the oil production until breakthrough is up to 45% higher when compared with water injection.

A bounded scheme based on Bézier curves for convection-dominated transport problems

This work presents a family of bounded schemes constructed based on the concept of Bézier curves that satisfies the TVD, CBC, and BAIR stability criteria for the numerical solution of convection-dominated transport problems in computational fluid dynamics problems. The convergence of the schemes to solve a series of 1D and 2D problems is analyzed. The numerical results of the proposed scheme are compared to other established methods from the literature. The numerical results show good agreement with exact solutions and numerical data from benchmarks. The findings of this work suggest the proposed family of methods as a promising alternative for solving convection-dominated problems.