Three-dimensional Flow Problems Research Articles

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Axial Green Function Method in an Efficient Projection Scheme for Incompressible Navier–Stokes Flow in Arbitrary Domains

We introduce a new approach to solving incompressible Navier–Stokes flow. This method combines a projection scheme with the Axial Green Function Method (AGM). Based on the Kim and Moin methods, our methodology employs a predictor–corrector mechanism to achieve stable and accurate velocity corrections. Using axial Green functions, we transform complex differential equations into simpler one-dimensional integral equations. These are strategically placed along a minimal axis-parallel lines, known as axial lines, within the flow domain. This transformation makes computation and analysis more efficient from a numerical viewpoint. A significant innovation in our approach is using one-dimensional axial Green functions tailored explicitly for the reaction–diffusion ordinary differential operator. These functions efficiently handle the discrete-time derivative and viscous terms of the momentum equation. Furthermore, our approach allows for the arbitrary construction of axis-parallel lines, facilitating analysis near critical flow regions and even enabling the random distribution of these lines. We validate our proposed method through numerical examples, demonstrating the convergence of numerical solutions, the effectiveness of arbitrarily constructed axis-parallel lines, and the potential extension of our method to three-dimensional flow problems. Additionally, this study provides a robust and adaptable alternative way to solve incompressible Navier–Stokes flows, proving its effectiveness in practical applications such as Tesla valves.

Surface tension and wetting at free surfaces in smoothed particle hydrodynamics

Surface tension and wetting are dominating physical effects in microscale and nanoscale flows. We present an efficient and reliable model of surface tension and equilibrium contact angles in smoothed particle hydrodynamics for free-surface problems. We demonstrate its robustness and accuracy by simulating several three-dimensional free-surface flow problems driven by interfacial tension.

A cell-based smoothed finite element model for the analysis of turbulent flow using realizable k-ε model and mixed meshes

Smoothed finite element method (S-FEM) has been widely employed in computational mechanics, particularly in computational solid mechanics (CSM) and, to a lesser extent, in computational fluid dynamics (CFD). The present work focuses on the development of S-FEM for solving three-dimensional incompressible turbulent flow problems using the realizable k-ε model. The Streamline Upwind Petrov-Galerkin approach combined with Stabilized Pressure Gradient Projection (SUPG/SPGP) was used to restrain/control the spatial oscillation and instability in conjunction with mixed meshes employed to discretize complex geometries. The presented turbulent cell-based S-FEM (CS-FEM) was validated under several common flow types, including mesh turbulence, turbulent channel flow, turbulent flow past a circular cylinder, flow separation around an obstacle, flow in the dimpled channel, and flow in the upper human airway. In particular, the presented work exhibited the latent capability of the CS-FEM in solving turbulent flows near boundary regions and strong convection regions. Meanwhile, the results of the CS-FEM were compared to those of the lowest equal-order finite element method (i.e. P1–P1 or Q1–Q1 elements) in simulating turbulent flows, and it was found that the CS-FEM had better agreement with other published works.

Validation of wall boundary conditions for simulating complex fluid flows via the Boltzmann equation: Momentum transport and skin friction

The influence and validity of wall boundary conditions for non-equilibrium fluid flows described by the Boltzmann equation remains an open problem. The substantial computational cost of directly solving the Boltzmann equation has limited the extent of numerical validation studies to simple, often two-dimensional, flow problems. Recent algorithmic advancements for the Boltzmann–Bhatnagar–Gross–Krook equation introduced by the authors [Dzanic et al., J. Comput. Phys. 486, 112146 (2023)], consisting of a highly efficient high-order spatial discretization augmented with a discretely conservative velocity model, have made it feasible to accurately simulate unsteady three-dimensional flow problems across both the rarefied and continuum regimes. This work presents a comprehensive evaluation and validation of wall boundary conditions across a variety of flow regimes, primarily for the purpose of exploring their effects on momentum transfer in the low Mach limit. Results are presented for a range of steady and unsteady wall-bounded flow problems across both the rarefied and continuum regimes, from canonical two-dimensional laminar flows to unsteady three-dimensional transitional and turbulent flows, the latter of which are the first instances of wall-bounded turbulent flows computed by directly solving the Boltzmann equation. We show that approximations of the molecular gas dynamics equations can accurately predict both non-equilibrium phenomena and complex hydrodynamic flow instabilities and show how spatial and velocity domain resolution affect the accuracy. The results indicate that an accurate approximation of particle transport (i.e., high spatial resolution) is significantly more important than particle collision (i.e., high velocity domain resolution) for predicting flow instabilities and momentum transfer consistent with that predicted by the hydrodynamic equations and that these effects can be computed accurately even with very few degrees of freedom in the velocity domain. These findings suggest that highly accurate spatial schemes (e.g., high-order schemes) are a promising approach for solving molecular gas dynamics for complex flows and that the direct solution of the Boltzmann equation can be performed at a reasonable cost when compared to hydrodynamic simulations at the same level of resolution.

Imposing slip conditions on curved boundaries for 3D incompressible flows with a very high-order accurate finite volume scheme on polygonal meshes

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.

Development of three-dimensional rotated lattice Boltzmann flux solver for the simulation of high-speed compressible flows

A three-dimensional (3D) Rotated Lattice Boltzmann Flux Solver (RLBFS) is proposed and analyzed with matrix-based linear stability theory for simulating compressible flows in a wide range of Mach numbers. The 3D RLBFS applies the finite volume method to discrete the Navier-Stokes equations and evaluates its fluxes at each cell interface. To improve numerical stability, the convective fluxes are obtained in a hybrid way by using the D1Q4 lattice Boltzmann model in two perpendicular directions decomposed by the outer normal vector of each cell interface. The viscous fluxes are computed in a conventional way using the second-order central scheme. The stability performance and order accuracy of the proposed 3D RLBFS are examined by using the matrix-based stability theory and L2 errors on different meshes respectively. It is shown that the 3D RLBFS is stable even at high Mach number and has the second-order accuracy in space. The reliability and capability of the proposed method is further evaluated by simulating several challenging compressible flow problems, including subsonic flow over the DLR-F4, transonic flow over the ONERA M6 wing, supersonic flow around NACA0012 wing, hypersonic flow over the Viking lander capsule, hypersonic flow over a hemisphere and hypersonic flow around a blunt nose double cone. The obtained numerical results are in good agreement with experimental and/or numerical data published in the literature, indicating that the present method provides a reliable and effective tool for simulating practical three-dimensional compressible flow problems in aeronautics and astronautics.

Stability of non-linear flow in heterogeneous porous media simulations using higher order and continuity basis functions

Using isogeometric analysis, we present a highly optimized FORTRAN code for simulations of three-dimensional non-linear flow problem in heterogeneous media. This includes deriving the weak formulations, discretizing with the forward Euler scheme, and implementing computation of the right-hand side. Some properties of exact solutions of heat transport and non-linear flow in heterogeneous media problems have been derived to validate the numerical solution. Stability analysis of the employed explicit time-stepping scheme has been performed. First, the properties of the discrete form of the heat transfer problem have been systematically investigated, leading to the conjecture concerning the sufficient condition of stability of the scheme, which numerical computations have subsequently confirmed. A similar analysis has been carried out for the non-linear flow in heterogeneous media problem. For this problem, the bound on the time step was conjectured to decrease exponentially, which has been verified experimentally. The simulation’s behavior close to the stability limit has been investigated and found to be considerably more complex than for the heat transport problem. Detailed experimental analysis of the time complexity of the particular parts of the simulator algorithm has been described. In addition, dependence on both mesh size and order of continuity of B-spline basis has been investigated.

Efficient dual-scale flow simulation for Resin Transfer Molding process based on domains skeletonization

In the Resin Transfer Molding (RTM) process, a polymeric resin is injected inside a dry preform to fill the gaps around and inside the fiber tows. Simulating this process at the scale of the tows is challenging because of the computational cost associated to solving a three-dimensional dual-scale flow problem. In this work, a novel Dual-Scale Skeleton model (DSS) is introduced, capable of solving a dual-scale flow problem at an affordable computational cost. The three-dimensional geometry of a multi-layer layup, consisting of inter-tow channels and permeable tows, is replaced by a skeletonized representation of the original subdomains. Dual-scale flow is modeled using a Reynolds-Darcy finite elements formulation. The model is validated numerically and its application is demonstrated over a few test cases. The adoption of the DSS model allows one to simulate complex dual-scale flow problems over large domains at a reduced computational cost when compared to full 3D solutions.

Modeling surface tension in Smoothed Particle Hydrodynamics using Young–Laplace pressure boundary condition

We introduce a surface tension model for Smoothed Particle Hydrodynamics (SPH) using a Young–Laplace pressure boundary condition. Our model does not rely on fitting parameters and can be applied to different variants of SPH. We demonstrate its robustness and accuracy by simulating several notoriously difficult three-dimensional free surface flow problems driven by surface tension.

New strategy for the numerical solution of multi-dimensional diffusion equations

PurposeThe purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is very easy to implement and handles the restrictive variable that may ruin the physical nature of the problem.Design/methodology/approachThis study combines Sawi transform (ST) and the homotopy perturbation method (HPM) to formulate the idea of Sawi homotopy perturbation transform method (SHPTM). First, this study implements ST to handle the recurrence relation and then incorporates HPM to derive the series solutions of this recurrence relation. ST has the advantage in that it does not require any assumptions or hypothesis for the evaluation of series solutions.FindingsThis strategy finds the results very accurate and close to the precise solution. The graphical observations and the surface solution demonstrate that SHPTM is a reliable and powerful scheme for finding the approximate solution of heat flow problems.Originality/valueThe study presents an original work. This study develops SHPTM for the approximate solution of two- and three-dimensional heat flow problems. The obtained results and graphical representation demonstrate that SHPTM is a very authentic and reliable approach.

A conservative finite element solver for the induction equation of resistive MHD: Vector potential method and constraint preconditioning

A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented. The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner. Simulation of three-dimensional driven cavity flow problem using full MHD solver is also conducted.

Family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method.

In this paper, we propose a family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method (FD-LBM), which consist of convex combinations. The FD-LBM unifies several different numerical schemes for the Navier-Stokes equations, and thereby these boundary schemes are naturally applicable for the standard LBM. The accuracy of the boundary schemes is independent of the boundary location, and it is validated by several numerical experiments, two- and three-dimensional flow problems, with straight and curved boundaries.

RETRACTED: Aqueous aluminium–copper hybrid nanofluid flow past a sinusoidal cylinder considering three-dimensional magnetic field and slip boundary condition

We analyzed the problem of the steady general three-dimensional magnetohydrodynamics stagnation-point boundary layer flow past an impermeable wavy circular cylinder considering aluminium–copper/water hybrid nanofluid as the working fluid and velocity slip as well as temperature jump boundary conditions. The induced magnetic field effect was also taken into account. The analytical procedure is based on the model that implements the nanoparticles and base fluid masses to formulate the equivalent volume fraction, equivalent density, and equivalent specific heat at constant pressure which is then substituted in the chosen single-phase thermophysical properties. Then, the foregoing relations were used in basic governing PDEs (partial differential equations), according to Tiwari–Das nanofluid scheme. It is worth mentioning that the bvp4c code from MATLAB software that is a famous finite-difference method has been exploited for solving the final similarity ODEs (ordinary differential equations). Results demonstrate that the developed mass-based model can be successfully employed with great confidence to study the flow and heat transfer of hybrid nanofluid in other similar problems. Moreover, it is proved that the nodal stagnation points possess higher values of skin friction coefficients and local Nusselt numbers relative to those for the saddle stagnation points. Besides, enhancing the second nanoparticle's mass leads to increase in all parameters of engineering interest including skin friction coefficients along x and y directions as well as local Nusselt number.

THREE-DIMENSIONAL STAGNATION-POINT FLOW OVER A PERMEABLE STRETCHING/SHRINKING SHEET WITH A SECOND ORDER SLIP FLOW

The problem of steady laminar three-dimensional stagnation-point flow on a permeable stretching/shrinking sheet with second order slip flow model is studied numerically. Similarity transformation has been used to reduce the governing system of nonlinear partial differential equations into the system of ordinary (similarity) differential equations. The transformed equations are then solved numerically using the \texttt{bvp4c} function in MATLAB. Multiple solutions are found for a certain range of the governing parameters. The effects of the governing parameters on the skin friction coefficients and the velocity profiles are presented and discussed. It is found that the second order slip flow model is necessary to predict the flow characteristics accurately.

A comprehensive and practical guide to the Hess and Smith constant source and dipole panel

ABSTRACT In a series of landmark reports and papers, J.L. Hess and A.M.O. Smith of Douglas Aircraft Company, Inc. introduced the quadrilateral constant source panel to solve three-dimensional nonlifting potential flow problems. Later a panel with constant dipole (doublet) distribution was added for lifting flow computations. Hess and Smith's publications provide equations for the computation of the velocities induced by the singularity distributions along with required geometric properties of the panel. Equations are presented considering an implementation in Fortran (Versions II and IV), the commonly used programming language for numerical methods at the time. The present paper builds on Hess and Smith's groundbreaking work, restating equations with modern programming languages in mind capable of fast vector operations like Fortran 95, Python or Julia. Formulas are provided for the computation of geometric properties, coordinate transformations, as well as first and second-order potential derivatives. Example input and output data allow readers to test and validate their own implementation.

Numerical evaluation of the latent heat thermal energy storage performance enhancement by installing longitudinal fins

Charging and discharging processes inside the shell-and-tube type latent heat thermal energy storage, which uses technical grade RT 25 paraffin as the phase change material and water as the heat transfer fluid, have been numerically investigated in order to evaluate the enhancement of the latent storage thermal performance when longitudinal fins are installed onto the tubes in comparison to the plain tube configuration. The study of latent storage tank performance enhancement using low thermal conductivity material, such as paraffin, is necessary to increase the efficiency of solar energy systems and reduce the operating cost of the system. Transient problem of three-dimensional fluid flow and heat transfer has been described with the defined mathematical model, in which simulation of phase change heat transfer is incorporated through the enthalpy formulation. Governing equations, with appropriate initial and boundary conditions, have been solved iteratively using the finite volume method in the ANSYS Fluent numerical solver using an upgraded numerical procedure. In order to validate the numerical procedure, numerical results have been compared with experimental measurements, obtained from experimental investigations carried out on the constructed shell-and-tube latent storage with longitudinal fins, and good agreement between numerical and experimental results is observed. The comparison of the two studied configurations shows a significant improvement in the overall heat transfer with the installation of fins in both the charging and discharging cycles, with a 52% reduction in the total melting time and a 43% reduction in the total solidification time compared to the plain tube configuration. Relative accumulated/released energy, defined as the ratio of accumulated/discharged energy and maximum storage capacity for the plain tube configuration, represents comprehensive latent storage efficiency and has been introduced as a performance enhancement metric. Finned tube latent storage is fully charged at relative accumulated energy value of 0.949, since maximum value of 1 can never be reached because of the decreased phase change material's mass due to fins. The same value is achieved 3 h later in the plain tube configuration. Similarly, finned tube latent storage is fully discharged at relative released energy value of 0.948, and the same value in plain tube configuration is reached 12 h later. Clearly, reduction in phase change material mass and obstruction of natural convection in the liquid phase due to installation of fins are negligible on the overall latent storage thermal performance during both charging and discharging. Results obtained by this investigation will be used as guidelines for further analyses focused on defining the optimal fin configuration and geometry parameters of shell-and-tube latent storage tanks.

A Semianalytical Model for Simulating Fluid Flow in Tight Sandstone Reservoirs with a Bottom Aquifer

Water breaks through along fractures is a major concern in tight sandstone reservoirs with a bottom aquifer. Analytical models fail to handle the three-dimensional two-phase flow problem for partially penetrating inclined fractures, so time-consuming numerical simulation are often used for this problem. This paper presents an efficient semianalytical model for this problem considering three-dimensional fractures and two-phase flow. In the model, the hydraulic fracture is handled discretely with a numerical discrete method. The three-dimensional volumetric source function in real space and superposition principle are employed to solve the model analytically for fluid flow in the reservoir. The transient flow equations for flow in three-dimensional inclined fractures are solved by the finite difference method numerically, in which two-phase flow and stress-dependent properties are considered. The eventual solution of the model and transient responses are obtained by coupling the model for flow in the reservoir and discrete fracture dynamically. The validation of the semianalytical model is demonstrated in comparison to the solution of the commercial reservoir simulator Eclipse. Based on the proposed model, the effects of some critical parameters on the characteristics of water and oil flow performances are analyzed. The results show that the fracture conductivity, fracture permeability modulus, inclination angle of fractures, aquifer size, perforation location, and wellbore pressure drop significantly affect production rate and water breakthrough time. Lower fracture conductivity and larger inclination angle can delay the water breakthrough time and enhance the production rate, but the increment tends to decline gradually. Furthermore, water breakthrough will occur earlier if the wellbore pressure drop and aquifer size are larger. Besides, the stress sensitivity and perforation location can delay the water breakthrough time.

Numerical investigation of three-dimensional nanofluid flow with heat and mass transfer on a nonlinearly stretching sheet using the barycentric functions

Purpose The purpose of this paper is to investigate a three-dimensional boundary layer flow with considering heat and mass transfer on a nonlinearly stretching sheet by using a novel operational-matrix-based method. Design/methodology/approach The partial differential equations that governing the problem are converted into the system of nonlinear ordinary differential equations (ODEs) with considering suitable similarity transformations. A direct numerical method based on the operational matrices of integration and product for the linear barycentric rational basic functions is used to solve the nonlinear system of ODEs. Findings Graphical and tabular results are provided to illustrate the effect of various parameters involved in the problem on the velocity profiles, temperature distribution, nanoparticle volume fraction, Nusselt and Sherwood number and skin friction coefficient. Comparison between the obtained results, numerical results based on the Maple's dsolve (type = numeric) command and previous existing results affirms the efficiency and accuracy of the proposed method. Originality/value The motivation of the present study is to provide an effective computational method based on the operational matrices of the barycentric cardinal functions for solving the problem of three-dimensional nanofluid flow with heat and mass transfer. The convergence analysis of the presented scheme is discussed. The benefit of the proposed method (PM) is that, without using any collocation points, the governing equations are converted to the system of algebraic equations.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.