Incompressible Navier-Stokes Research Articles

A multiblock (MIB) finite element method for accurate and efficient blood flow simulation

We present an efficient and accurate immersed boundary (IB) finite element (FE) method for internal flow problems with complex geometries (e.g. blood flow in the vascular system). In this study, we use a voxelized flow domain (discretized with hexahedral and tetrahedral elements) instead of a box domain, which is frequently used in IB methods. We highlight the applicability and efficiency of the proposed method in generating the flow domain mesh and removing unnecessary portions of the domain, compared to the standard IB methods that rely on a full Cartesian mesh. Our approach is attractive for internal flow problems that use IB methods. The proposed method accurately satisfies the boundary conditions on the immersed object using velocity and pressure correction procedures. The velocity correction applies implicitly such that the velocity on the immersed boundary (Lagrangian points) interpolated from the corrected velocity values computed on the mesh nodes (Eulerian nodes) accurately satisfies the prescribed velocity boundary conditions. The proposed method utilizes the well-established incremental pressure correction scheme (IPCS) FE solver, and the boundary condition-enforced IB (BCE-IB) method to numerically solve the transient, incompressible Navier-Stokes (NS) flow equations. We verify the accuracy of our numerical method using the analytical solution for the Poiseuille flow in a cylinder, and the available experimental data (laser Doppler velocimetry) for the flow in a three-dimensional 90° angle tube bend. We further examine the accuracy and applicability of the proposed method by considering flow within complex geometries, such as blood flow in aneurysmal vessels and the aorta, flow configurations that would otherwise be difficult to solve by most IB methods. Our method offers high accuracy, as demonstrated by the verification examples, and high applicability, as demonstrated through the solution of blood flow within complex geometry. The proposed method is efficient, since it is as fast as the traditional finite element method used to solve the Navier–Stokes flow equations, with a small overhead (not more than 5%) due to the numerical solution of a linear system formulated for the IB method.

A three-dimensional non-orthogonal multiple-relaxation-time phase-field lattice Boltzmann model for multiphase flows at large density ratios and high Reynolds numbers

This study proposes a three-dimensional non-orthogonal multiple-relaxation-time (NMRT) phase-field multiphase lattice Boltzmann (PFLB) model within a recently established unified lattice Boltzmann model (ULBM) framework [Luo et al., Phil. Trans. R. Soc. A 379, 20200397, 2021]. The conservative Allen-Cahn equation and the incompressible Navier-Stokes (NS) equations are solved. In addition, a local gradient calculation scheme for the order parameter of the Allen-Cahn equation is constructed with the non-equilibrium part of the distribution function. A series of benchmark cases are conducted to validate the proposed model, including the two-phase Poiseuille flow, Rayleigh-Taylor instability, binary liquid/metal droplet collision, and a bubble rise in water. The present simulation results are in good agreement with existing simulation and experimental data. In the simulation of the co-current two-phase Poiseuille flow, the present model is proven to resolve the discontinuity at the phase interface and provide accurate results at extremely high density ratios (i.e., up to 106). Finally, the proposed model is adopted to simulate two challenging cases: (1) water droplet splashing during its impacting on a thin liquid film and (2) liquid jet breakup. The simulation results demonstrate an excellent agreement with previous experimental results, both qualitatively and quantitatively. In these simulations, the Weber number and Reynolds number reach 105 and 6000, respectively, and the viscosity can be as low as ∼10−4, in the lattice unit.

A physics-preserving pure streamfunction formulation and high-order compact solver with high-resolution for three-dimensional steady incompressible flows

In this paper, a pure streamfunction high-order compact (HOC) difference solver is proposed for three-dimensional (3D) steady incompressible flows. A physics-preserving pure streamfunction formulation is first introduced for the steady 3D incompressible Navier–Stokes (NS) equations without in-flow and out-flow boundary conditions, where the divergence of streamfunction ∇ · ψ is maintained in the convective and the vortex-stretching terms together in the nonlinear term of equations to reduce the physics-informed loss. Moreover, taking the streamfunction-vector components and their first-order partial derivatives as unknown variables, some fourth-order compact schemes are suggested for the partial derivatives that appear in the streamfunction formulation, and a high-resolution HOC scheme is introduced for approximating the pure third-order partial derivatives in the convective term. Meanwhile, a new HOC scheme is proposed for the first-type boundary conditions of the streamfunction. Finally, a fourth-order compact difference scheme and its algorithm are established for the 3D steady incompressible NS equations in the streamfunction form, subject to no in-flow and out-flow boundary conditions. Several numerical examples are carried out to validate and prove the accuracy, convergence, and efficiency of the proposed new method. Numerical results reveal that the proposed method not only can achieve fourth-order accuracy but also has excellent convergence, high-resolution, and low computational cost at higher Reynolds number.

A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows

This review paper aims to summarize recent advancements in time-marching schemes for solving Navier–Stokes (NS) equations in multiphase flow simulations. The focus is on dual-time stepping, local preconditioning, and artificial compressibility methods. These methods have proven to be effective in achieving high time accuracy in simulations, as well as converting the incompressible NS equations into a hyperbolic form that can be solved using compact schemes, thereby accelerating the solution convergence and allowing for the simulation of compressible flows at all Mach numbers. The literature on these methods continues to grow, providing a deeper understanding of the underlying physical processes and supporting technological advancements. This paper also highlights the imposition of dual-time stepping on both incompressible and compressible NS equations. This paper provides an updated overview of advanced methods for the CFD community to continue developing methods and select the most suitable two-phase flow solver for their respective applications.

Immiscible two-phase model for air blasts created during natural avalanches

An immiscible two-phase model based on the incompressible Navier-Stokes (N-S) equations is used to simulate the air blast generated by an avalanche. For simplicity, the avalanche is treated as an assembly of monodisperse spherical grains and described as a continuous media. The constitutive law of local µ(I) rheology is introduced to model the moving granular material. The motion of the avalanche and the induced air blast fits into a unified framework that combines the N-S−type governing equations with a µ(I)-rheology−based kinematic viscosity and a constant viscosity. The avalanche-air interface is treated using the volume-of-fluid method. A numerical program was developed on the open-source platform OpenFOAM specifically for this model to simulate the entire evolutionary process of the avalanche as well as the air blast generated. The model was validated by comparing the results of numerical simulations with those from inclined-plane laboratory experiments. With terrain input from the Shuttle Radar Topography Mission data, the model was further applied to simulate the air blast generated in two natural avalanches, namely, the Baige and Wenjia valley avalanches fo China, which occurred in 2008 and 2018, respectively. The simulation results were found to be consistent with field observations following a statistical analysis of the properties of the air blast including flow speed and area of impact of the above-mentioned natural events.

HIFIR: Hybrid Incomplete Factorization with Iterative Refinement for Preconditioning Ill-Conditioned and Singular Systems

We introduce a software package called Hybrid Incomplete Factorization with Iterative Refinement (HIFIR) for preconditioning sparse, unsymmetric, ill-conditioned, and potentially singular systems. HIFIR computes a hybrid incomplete factorization (HIF) , which combines multilevel incomplete LU factorization with a truncated, rank-revealing QR (RRQR) factorization on the final Schur complement. This novel hybridization is based on the new theory of ϵ-accurate approximate generalized inverse (AGI) . It enables near-optimal preconditioners for consistent systems and enables flexible GMRES to solve inconsistent systems when coupled with iterative refinement. In this article, we focus on some practical algorithmic and software issues of HIFIR. In particular, we introduce a new inverse-based rook pivoting (IBRP) into ILU, which improves the robustness and the overall efficiency for some ill-conditioned systems by significantly reducing the size of the final Schur complement for some systems. We also describe the software design of HIFIR in terms of its efficient data structures for supporting rook pivoting in a multilevel setting, its template-based generic programming interfaces for mixed-precision real and complex values in C++, and its user-friendly high-level interfaces in MATLAB and Python. We demonstrate the effectiveness of HIFIR for ill-conditioned or singular systems arising from several applications, including the Helmholtz equation, linear elasticity, stationary incompressible Navier–Stokes (INS) equations, and time-dependent advection-diffusion equation.

LES/DNS fluid-structure interaction simulation of non-linear slender structures in Nektar++ framework

Nektar++ is a spectral/hp element open-source framework written in C++ for the construction of classical low-order h-type as well as higher-order p-type finite element solvers. It seeks to overcome the implementation challenges of the complex data structures associated with high-order finite element methods; hence, providing an efficient, flexible and HPC scalable platform for the development of solvers for partial differential equations using the spectral/hp element method. In the present work, capabilities of Nektar++ is leveraged for development of two fluid-structure interaction (FSI) solvers for simulations of highly deformable nonlinear slender structures. The FSI solver uses the incompressible Navier-Stokes (NS) solvers of Nektar++ for fluid flow while the structural dynamics is modelled using Geometrically-Exact Composite Beams (GECB). The open-source SHARPy framework is linked to Nektar++ and used for structural simulation. Aiming at high-fidelity (LES/DNS) FSI simulations, the thick-strip approach is used to reduce computational costs. In this approach, the full 3D fluid domain is represented with series of smaller 3D domains normal to the local axis of the structure and having a finite thickness in the spanwise direction where periodicity is also assumed. Hence, while reducing the computational costs by avoiding the discretization of equations over the entire slender structure, the strip thickness allows capturing the local 3D turbulent wake and accurately predict the fluid forces on the structure. Two approaches are adopted to avoid the dynamic remeshing due to the large and non-linear deformation of the structure. In the first approach, the transformed the NS equations are solved in the non-inertial body-fitted coordinates while in the second approach the NS equations are formulated in the moving frame of reference and solved with the spectral/hp element method. A hybrid parallelisation approach of Nektar++ is extended for the thick-strip method which allows having non-constant cross-section along the structural span as well as efficient and flexible use of computational resources, and excellent HPC performance for the FSI simulations. The capability of the FSI solver is demonstrated via several examples.

Analysis of buoyancy driven flow inside a vertical filter chamber

Designs based on scientific evidence play an essential role when engineering machinery. The current case study seeks an alternative way of filtering particles using a permeable surface. The filter is designed so that the effects of heat from a permeable bottom wall create a force that drives permeates out of the filter chamber. The model follows the mass conservation, incompressible Navier-Stokes and the energy conservation equations. Effects of parameters arising from flow dynamics and heat distribution are studied to better understand the theory behind the filtrations process. Graphical representations of skin friction and Nusselt number inside the filter chamber are plotted against filtration dynamics parameters. We found that more permeates are produced when the internal energy is low since the system loses internal temperature when less dense fluid, which possesses heat, is withdrawn from the chamber. In addition, the chamber volume increase permeates outflow since it enhances the pulling effects of the buoyancy force. It is found that increasing the Richardson number enhances the buoyancy forces and viscous effects; thus, decreasing permeates outflow velocity and eventually reduces wall drag coefficient. The wall drag coefficients on the left and right vertical walls have extrema towards the bottom of the filter chamber, in the region, 0 <y < 2 % H.

Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping

Abstract The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε → 0 + \varepsilon \to {0}^{+} on the two-dimensional space.

Filtering efficiency model that includes the statistical randomness of non-woven fiber layers in facemasks.

Facemasks have become important tools to fight virus spread during the recent COVID-19 pandemic, but their effectiveness is still under debate. We present a computational model to predict the filtering efficiency of an N95-facemask, consisting of three non-woven fiber layers with different particle capturing mechanisms. Parameters such as fiber layer thickness, diameter distribution, and packing density are used to construct two-dimensional cross-sectional geometries. An essential and novel element is that the polydisperse fibers are positioned randomly within a simulation domain, and that the simulation is repeated with different random configurations. This strategy is thought to give a more realistic view of practical facemasks compared to existing analytical models that mostly assume homogeneous fiber beds of monodisperse fibers. The incompressible Navier-Stokes and continuity equations are used to solve the velocity field for various droplet-laden air inflow velocities. Droplet diameters are ranging from 10nm to 1.0µm, which covers the size range from the SARS-CoV-2 virus to the large virus-laden airborne droplets. Air inflow velocities varying between 0.1m·s-1 to 10m·s-1 are considered, which are typically encountered during expiratory events like breathing, talking, and coughing. The presented model elucidates the different capturing efficiencies (i.e., mechanical and electrostatic filtering) of droplets as a function of their diameter and air inflow velocity. Simulation results are compared to analytical models and particularly compare well with experimental results from literature. Our numerical approach will be helpful in finding new directions for anti-viral facemask optimization.

A method of immersed layers on Cartesian grids, with application to incompressible flows

The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and derived forms such as the projection method of Taira and Colonius (J. Comput. Phys., 2007), have been useful for simulating flow physics in problems with moving interfaces on stationary grids. However, in their interface treatment, these methods do not distinguish one side from the other, but rather, apply the motion constraint to both sides, and the associated interface force is an inseparable mix of contributions from each side. In this work, we define a discrete Heaviside function, a natural companion to the familiar discrete Dirac delta function (DDF), to define a masked version of each field on the grid which, to within the error of the DDF, takes the intended value of the field on the respective sides of the interface. From this foundation we develop discrete operators and identities that are uniformly applicable to any surface geometry. We use these to develop extended forms of prototypical partial differential equations, including Poisson, convection-diffusion, and incompressible Navier-Stokes, that govern the discrete masked fields. These equations contain the familiar forcing term of the IBM, but also additional terms that regularize the jumps in field quantities onto the grid and enable us to individually specify the constraints on field behavior on each side of the interface. Drawing the connection between these terms and the layer potentials in elliptic problems, we refer to them generically as immersed layers. We demonstrate the application of the method to several representative problems, including two-dimensional incompressible flows inside a rotating cylinder and external to a rotating square.

An artificial Equation of State based Riemann solver for a discontinuous Galerkin discretization of the incompressible Navier–Stokes equations

In this work we present a novel exact Riemann solver for an artificial Equation of State (EoS) based modification of the incompressible Euler equations. Differently from the well known artificial compressibility method, this new approach overcomes the lack of the pressure-velocity coupling by using a suitably designed artificial EoS. The modified set of equations fits into the framework of first order hyperbolic conservation laws. Accordingly, an exact Riemann solver is derived. This new solver has the advantage to avoid a wave pattern violation issue which can affect the exact Riemann problem solution obtained by the standard artificial compressibility approach. The new artificial EoS based Riemann solver can be used as a tool for the definition of the advective Godunov fluxes in a Finite Volume or a discontinuous Galerkin discretization of the incompressible Navier–Stokes (INS) equations. We assess and analyse the new exact Riemann solver on 1D Riemann problems. Its capability and effectiveness are then shown in the context of a high-order accurate discontinuous Galerkin discretization of the INS equations. In particular, we verify the convergence properties, both in space and time, considering two test cases in 2D, namely, the Kovasznay test case and a damped travelling waves test case. Finally, we perform an implicit large eddy simulation of the incompressible turbulent flow over periodic hills where the classic forcing term formulation is modified to deal with variable time steps. Solution comparisons with respect to numerical and experimental results from the literature are given.

Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the Mach number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

Evaluation of color image interpolation based on incompressible Navier Stokes technique

Color image interpolation encompasses reconstructing parts of a video or an image based on information from the neighbor. Technique involves the restoration of noised photos and animation or image denoising. The Navier-Stokes (NS) technique has been widely investigated as an essential research by image restoration. These NS equations contribute spectacular results for producing an animation as they augment reality. They can boost real-time video games to be more sensible than ever. In this paper, we present the Incompressible NS approach (INS) for color image interpolating. The method per se is based on fluid flow concept to circulate directed lines from the peripheral into the area to be interpolated. The image intensity represents stream function in a computational flow of 2D fluid dynamics. The algorithm is implemented to carry on lines regarding gradient vectors at the edge of the interpolating region. It uses the improvement of powerful numerical analysis. It is also proven as an innovative idea for easing problems in image analytics as well as computer vision.

Self-propelled motion of a rigid body inside a density dependent incompressible fluid

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole ℝ3. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.

A Ffowcs Williams-Hawkings numerical aeroacoustic study of varied and fixed-pitch blades of an H-Rotor vertical axis wind turbine

The effects of sound pressure level at two observation positions of a fixed and varied blade pitch angle at Low-Mach unsteady incompressible Reynolds-Average Navier-Stokes flow approach, on an H-rotor Vertical Axis Wind Turbine. The objective of the study is to compare the noise dissipation and output power/energy of the airfoil blades design of the vertical axis wind turbine in residential zones. The Ffowcs Williams-Hawkings (FHWH) techniques were applied to validate the output noise and vortex shedding of the different angles of attacks (AoA). The study postulated that the time history of the calculated sound pressure level at two observers positions: the aeroacoustic, blade vortex interaction noise, flow separations, dynamic stall experience from varied angled of attacks are found to produces less noise and vortex shedding compared to the fixed angle of attack.

STUDY OF THE BEHAVIOURS OF SINGLE-PHASE TURBULENT FLOW AT LOW TO MODERATE REYNOLDS NUMBERS THROUGH A VERTICAL PIPE. PART I: 2D COUNTERS ANALYSIS

&#x0D; This study presents a model to investigate the behavior of the single-phase turbulent flow at low to moderate Reynolds number of water through the vertical pipe through (2D) contour analysis. The model constructed based on governing equations of an incompressible Reynolds Average Navier-Stokes (RANS) model with (k-ε) method to observe the parametric determinations such as velocity profile, static pressure profile, turbulent kinetic energy consumption, and turbulence shear wall flows. The water is used with three velocities values obtained of (0.087, 0.105, and 0.123 m/s) to represent turbulent flow under low to moderate Reynolds number of the pipe geometry of (1 m) length with a (50.8 mm) inner diameter. The water motion behavior inside the pipe shows by using [COMSOL Multiphysics 5.4 and FLUENT 16.1] Software. It is concluded that the single-phase laminar flow of a low velocity, but obtained a higher shearing force; while the turbulent flow of higher fluid velocity but obtained the rate of dissipation of shearing force is lower than that for laminar flow. The entrance mixing length is affected directly with pattern of fluid flow. At any increasing in fluid velocity, the entrance mixing length is increase too, due to of fluid kinetic viscosity changes. The results presented the trends of parametric determinations variation through the (2D) counters analysis of the numerical model. When fluid velocity increased, the shearing force affected directly on the layer near-wall pipe. This leads to static pressure decreases with an increase in fluid velocities. While the momentum changed could be played interaction rules between the fluid layers near the wall pipe with inner pipe wall. Finally, the agreement between present results with the previous study of [1] is satisfied with the trend&#x0D;

A magnetic field coupling lattice Boltzmann model and its application on the merging process of multiple-ferrofluid-droplet system

The present study concerns the dynamics of ferrofluid droplets in a Newtonian fluid. The incompressible Navier-Stokes (NS) equations are applied for the dynamics of the immiscible magnetic and Newtonian fluids while the Cahn-Hilliard (C-H) equation is adopted to describe the behavior of their interface. A magnetic field coupling lattice Boltzmann (LB) model is developed to solve both the NS equations and the C-H equation. Specially, a mass-correcting term is introduced into the C-H equation, which strongly enforces the mass conservation. The magnetic field is evaluated by a Poisson equation solver with a self-correcting procedure for the static Maxwell equations. The magnetic dipole force is transformed into the magnetic surface force by a rigorous mathematical procedure, which can physically describe the magnetic effect on the interface. Moreover, the magnetic force after this treatment becomes easy-to-implement, which can be directly incorporated into the external force term of the LB model. The capability of the present combination method to simulate the magnetic multiphase flows is demonstrated by three typical numerical examples, i.e., Laplace law for a stationary droplet, a stationary cylinder under an external uniform magnetic field, the deformation of a single ferrofluid droplet. Further, the merging process of multiple-ferrofluid-droplet system in organic oil is investigated. It is found that the elongation of ferrofluid droplet in the direction of magnetic field shows positive impact on the merging process when the orientation of ferrofluid droplets is parallel to the external magnetic field, while negative impact on the merging process when their orientation is perpendicular to the external magnetic field.

An energy-stable scheme for a 2D simple fluid-particle interaction problem

We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different time steps. In the proposed numerical scheme, the tALE mesh velocity, the incompressible NS equations, and Newton's second law are solved simultaneously. We prove that under certain conditions, the new time discretization scheme satisfies an energy law. For the space discretization, the extended finite element method (XFEM) is used to solve the problem on a fixed Cartesian mesh. The developed method is first-order accurate in time and space without being momentum conservative. To verify the accuracy and stability of our numerical scheme, we present numerical experiments including the fitting of the Jeffery orbit by rotating of an ellipse and the free-falling of an elliptic particle in water.

Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport

The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.