Simulation Of Incompressible Viscous Flows Research Articles

Efficiency of local Vanka smoother geometric multigrid preconditioning for space‐time finite element methods to the Navier–Stokes equations

AbstractNumerical simulation of incompressible viscous flow continues to remain a challenging task, in particular if three space dimensions are involved. Space‐time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Using a temporal test basis supported on the subintervals and linearizing the resulting algebraic problems by Newton's method yield linear systems of equations with block matrices built of (k + 1) × (k + 1) saddle point systems, where k denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a V‐cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two‐ and three‐dimensional benchmark problem of flow around a cylinder. The robustness of the solver with respect to the piecewise polynomial orderkin time is analyzed and confirmed numerically.

On Shadowing Effects in the Flow around the Group of Circular Cylinders in the Geotechnical Applications

The purpose of the presented study is to investigate the influence of the distance between three infinite circular cylinders in a tandem group on their aerodynamic coefficients and the general flow pattern. The questions of the numerical simulation of viscous incompressible flow around a group of cylinders in the tandem arrangement are considered. The numerical study of the flow around the group of cylinders is based on unsteady Reynolds-averaged Navier-Stokes equations. The results are presented as a visualization of the instantaneous flow parameters and the dependencies of the averaged drag coefficients on the distance between the cylinders. Following the previous investigations, the flow parameters were chosen in a little-studied range, corresponding to the Reynolds number of 120 for a standalone cylinder. The distances between the centers of neighboring cylinders ranged from 1.1 to 6.0 cylinder diameters. Three characteristic flow patterns are distinguished depending on the distance between the cylinders. The first one is a steady-state flow similar to the flow around the standalone cylinder. This flow pattern is formed when the distance between the cylinders is less than three diameters. Increasing the distance between the centers up to four diameters produces the second flow pattern. The vortex street is formed behind the group while the flow between the cylinders remains steady. Further increasing the distance between the cylinders causes the formation of the third pattern when the flow between the cylinders becomes unsteady as well. The averaged drag coefficients for each cylinder in the group are obtained. The presented work helps better to understand the flow structure around the group of cylinders and can be used at the design stage in various engineering applications, including cooling towers, heat exchanger tubes, chimney stacks, marine platforms, and high-rise buildings.

An implicit Cartesian cut-cell method for incompressible viscous flows with complex geometries

A versatile conservative three-dimensional Cartesian cut-cell method for simulation of incompressible viscous flows over complex geometries is presented in this paper. The present method is based on the finite volume method on a non-uniform staggered grid together with a consistent mass and momentum flux computation. Contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which avoids numerical instability without any additional small cut-cell treatment. Strict conservation of the mass and momentum for both fluid and cut cells is enforced through the PISO algorithm for the pressure–velocity coupling. The versatility and robustness of the present cut-cell method are demonstrated by simulating various two- and three-dimensional canonical benchmarks (flow over a circular cylinder, airfoil, sphere, pipe, and heart sculpture) and the computed results agree well with previous experimental measurements and various numerical results obtained from the boundary-fitted, immersed boundary/interface, and other cut-cell methods, verifying the accuracy of the proposed method.

Pore-scale modelling of fluid flow in porous media using the projection method for incompressible Navier-Stokes equations in irregular domains

This paper presents the results of numerical simulation of incompressible viscous flow in porous media, which comprise periodically arranged cylinders. This simulation is based on the numerical solution of the incompressible Navier-Stokes equations in irregular domains using the projection method on staggered grids, where the irregular boundary is represented by its level-set function at the pore-scale level. The main problem in numerical calculation of fluid flow through porous media occurs when the value of the porosity is close to 1 or is close to the threshold value since it is necessary to take a very fine numerical mesh, which requires additional computing power and increases the calculation time. There are exact analytical solutions for simple types of porous media which consist of periodically arranged cylinders. In this paper, the permeabilities of these porous media were numerically calculated and compared with the previous works based on the numerical solution of the Lattice-Boltzmann equation in irregular domains, when the fluid flow obeys Darcy’s law. The comparison of numerical and theoretical values of porosity shows that this method is sufficiently accurate for porosity values φ=0.2–0.8.

A Model for Stokes Flow in Domains with Permeable Boundaries

We derive a new computational model for the simulation of viscous incompressible flows bounded by a thin, flexible, porous membrane. Our approach is grid-free and models the boundary forces with regularized Stokeslets. The flow across the porous membranes is modeled with regularized source doublets based on the notion that the flux velocity across the boundary can be viewed as the flow induced by a fluid source/sink pair with the sink on the high-pressure side of the boundary and magnitude proportional to the pressure difference across the membrane. Several validation examples are presented that illustrate how to calibrate the parameters in the model. We present an example consisting of flow in a closed domain that loses volume due to the fluid flux across the permeable boundary. We also present applications of the method to flow inside a channel of fixed geometry where sections of the boundary are permeable. The final example is a biological application of flow in a capillary with porous walls and a protein concentration advected and diffused in the fluid. In this case, the protein concentration modifies the pressure in the flow, producing dynamic changes to the flux across the walls. For this example, the proposed method is combined with finite differences for the concentration field.

Simplified method for simulation of incompressible viscous flows inspired by the lattice Boltzmann method.

The lattice Boltzmann method (LBM) has gained increasing popularity in incompressible viscous flow simulations, but it uses many distribution functions (far more than the flow variables) and is often memory demanding. This disadvantage was overcome by a recent approach that solves the more actual macroscopic equations obtained through Taylor series expansion analysis of the lattice Boltzmann equations [Lu etal., J. Comput. Phys. 415, 109546 (2020)JCTPAH0021-999110.1016/j.jcp.2020.109546]. The key is to keep some small additional terms (SATs) to stabilize the numerical solution of the weakly compressible Navier-Stokes equations. However, there are many SATs that complicate the implementation of their method. Based on some analyses and numerous tests, we ultimately pinpoint two essential ingredients for stable simulations: (1) suitable density (pressure) diffusion added to the continuity equation and (2) proper numerical dissipation related to the velocity divergence added to the momentum equations. Then we propose a simplified method that is not only easier to implement but noticeably faster than the original method and the LBM. It contains much simpler SATs that only involve the density (pressure) derivatives, and it requires no intermediate steps or variables. As well, it is extended for thermal flows with small temperature variations and for two-phase flows with uniform density and viscosity. Several test cases, including some two-phase problems under two-dimensional, axisymmetric, and three-dimensional geometries, are presented to demonstrate its capability. This work may help pave the way for the simplest simulation of incompressible viscous flows on collocated grids based on the artificial compressibility methodology.

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of degree $N$ and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree $N$ on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadrature-free and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.

Direct numerical simulation of viscous incompressible flow with spherical particles in the flat channel

The paper deals with numerical simulation of 3D fluid flow with relatively large spherical particles in the flat channel. The simulation is performed using the previously developed SIMPLE based immersed boundary method. The collision model, describing particle-particle and particle-wall collisions, complements the numerical method. This model prevents particle overlapping and takes into account the complicated fluid flow in the space between the particles. The computed flow rates decrease significantly with increasing the particle concentration. It is shown that the value of the particle diameter has a small influence on the flow rate. The flow rate can be approximated using the value of effective viscosity computed by Maron – Peirce formula with the use of extremely high packing concentration. Particles are moving faster than the mixture, and the difference increases with increasing concentration.

Grid-Free Modelling Based on the Finite Particle Method for Incompressible Viscous Flow Problems

In this paper, we present a grid-free modelling based on the finite particle method for the numerical simulation of incompressible viscous flows. Numerical methods of interest are meshless Lagrangian finite point scheme by the application of the projection method for the incompressibility of the Navier–Stokes flow equations. The moving least squares method is introduced for approximating spatial derivatives in a meshless context. The pressure Poisson equation with Neumann boundary condition is solved by the finite particle method in which the fluid domain is discretized by a finite number of particles. Also, a continuous particle management has to be done to prevent particles from moving into configurations problematic for a numerical approximation. With the proposed finite particle technique, problems associated with the viscous free surface flow which contains the study on the liquid sloshing in tanks with low volumetric fluid type, solitary waves movement, and interaction with a vertical wall in numerical flume as well as the vortex patterns of the ship rolling damping are circumvented. These numerical models are investigated to validate the presented grid-free methodology. The results have revealed the efficiency and stability of the finite particle method which could be well handled with the incompressible viscous flow problems.

Estimating lift from wake velocity data in flapping flight

The application of the Kutta–Joukowski (KJ) theorem to estimating the lift of a flying animal based on wake velocity fields often leads to significant underprediction of the lift, which is known as the wake momentum paradox. This work attempts to answer the puzzling question on whether the KJ theorem is legitimate in its use for complex viscous unsteady wakes generated by flapping wings. The limitations in applying the KJ theorem to flapping wings are quantitatively examined through numerical simulations of viscous incompressible flows over three flapping wing models. The three flapping wing models studied in this work are a flapping wing with a fixed wingspan, a flapping wing with a dynamically changing wingspan and a dihedral flapping wing. The KJ theorem fails to give a satisfactory prediction of the time-averaged lift unless an effective span length is correctly computed. We propose a wake-sectional Kutta–Joukowski (WS-KJ) model to predict the time-averaged lift, where the effective span length is computed based on the time-averaged distance between the streamwise vorticity centroids in the right and left half sides of the Trefftz plane. The WS-KJ model incorporates the spatial evolutionary effects of the complex vortex structures in the wake and significantly improves the prediction of the time-averaged lift. The physical foundation for such improvement is explored. In addition, the time-dependent amplitude and phase changes of the unsteady lift are discussed as the fluid acceleration effect.

A boundary thickening-based direct forcing immersed boundary method for fully resolved simulation of particle-laden flows

A boundary thickening-based direct forcing (BTDF) immersed boundary (IB) method is proposed for fully resolved simulation of incompressible viscous flows laden with finite size particles. By slightly thickening the boundary thickness, the local communications between the Lagrangian points on the solid boundary and their neighboring fluid Eulerian grids are improved, based on an implicit direct forcing (IDF) approach and a partition-of-unity condition for the regularized delta function. This strategy yields a simple, yet much better imposition of the non-slip and non-penetration boundary conditions than the conventional direct forcing (DF) technique. In particular, the present BTDF method can achieve a numerical accuracy comparable with other representative improved methods, such as multi-direct forcing (MDF), implicit velocity correction (IVC) and the reproducing kernel particle method (RKPM), while its computing cost remains much lower than them and nearly equivalent to the conventional DF method. The dependence of the optimum thickness value of boundary thickening on the form of the different regularized delta functions is also revealed. By coupling the lattice Boltzmann method (LBM) with BTDF-IB, the robustness of the present BTDF IB method is demonstrated using numerical simulations of creeping flow (Re=0.1), steady vortex separating flow (Re=40) and unsteady vortex shedding flow (Re=200) around a circular cylinder. The accuracy and robustness of the present method for moving particle-laden flows are performed on three benchmark cases, such as free sedimentation of single circular cylinder, DKT sedimentation of two particles and Rayleigh-Taylor instability of 504 particles sedimentation in an enclosed cavity. The computational efficiency between the present BTDF method and the MDF method is also compared at last.

Construction of knotted vortex tubes with the writhe-dependent helicity

We propose a feasible method for constructing knotted vortex tubes with the finite thickness and arbitrary complexity and develop an accurate algorithm to implement this method in numerical simulations. The central axis of the knotted vortex tube is determined by the parametric equation of a given smooth and non-degenerate closed curve. The helicity of the vortex tube is only proportional to the writhe of the vortex axis, a geometric measure for coiling of vortex tubes. This vortex construction can facilitate the investigation of the conversion of writhe to twist in the helicity evolution of knotted vortex tubes. As examples, we construct velocity–vorticity fields of trefoil, cinquefoil, and septafoil vortex knots. These vortex knots are used as initial conditions in the direct numerical simulation of viscous incompressible flows in a periodic box. In the evolution of vortex knots from simple flows to turbulent-like flows, all the knots are first untied. Then the vortex topology is invariant and the helicity is almost conserved for the trefoil knot, whereas the helicity decays rapidly during the breakdown and coaxial interactions of pinch-off vortex rings for cinquefoil and septafoil knots.

An Overview of Numerical Methods for Incompressible Viscous Flow with Moving Particles

The importance of the fluid–particle interaction problem is considerable. Tremendous attention has been paid by researchers in the past two decades. This paper presents an overview of the representative methods for the simulation of incompressible viscous flow with moving boundaries based on conforming or nonconforming meshes. These methods include the distributed Lagrange multiplier/fictitious domain method, the immersed boundary method, the lattice Boltzmann method, and the arbitrary Lagrange–Eulerian method. The classification of the selected methodology is made to create a firm foundation for the future. The latest developments in the field during past few years are also reviewed. The opportunities and limits of all the methods are discussed to improve the research of particulate flow. Finally, the paper points out several directions of future researches.

A reconstruction-based cell-centered high-order finite volume method for incompressible viscous flow simulation on unstructured meshes

A new high-order (> 2nd order) cell-centered finite volume method is presented for incompressible flow simulation on unstructured meshes. Artificial compressibility is employed to couple the continuity and momentum equations in a manner that allows them to be solved simultaneously. A new numerical stencil, a so-called wrapping stencil, is utilized for linear and quadratic solution reconstruction in order to achieve more accurate and robust solution reconstruction not only for the interior cells, but also for the cells on the boundary, where fewer neighboring cells typically exist. The effectiveness of the current algorithm is demonstrated by various test cases, including an analytical solution reconstruction test, Kovasznay flow simulations with various Reynolds numbers, a driven cavity flow, and flow past a square cylinder. Based on the comparison with the standard low order scheme, the proposed second and third order schemes, based on linear and quadratic solution reconstruction, show superior accuracy, which sheds light on the method's applicability in solving more challenging incompressible flow problems.

Towards Multidimensional Artificially Characteristic-Based Scheme for Incompressible Thermo-Fluid Problems

A new multidimensional artificially characteristic-based (MACB) scheme is presented for simulation of incompressible viscous flows with heat transfer. Multidimensional characteristic relations for energy propagation in incompressible flow are derived for the first time. According to the new scheme, two-dimensional flow with heat transfer in a square cavity and forced convection around a circular cylinder are solved for a wide range of Reynolds and Grashof numbers. It was found a remarkable faster convergence of MACB in comparison with conventional CB scheme. The computed results of MACB scheme are in good agreement with the benchmark solutions. DOI: http://dx.doi.org/10.5755/j01.mech.23.6.15804

Moving immersed boundary method

SummaryA novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no‐slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier‐Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second‐order accurate projection method as a plug‐in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.

An ALE particle method using upwind interpolation

Abstract In the present paper, an ALE particle method is presented for the numerical simulation of incompressible viscous flows. The present approach interpolates physical quantities, e.g., velocity and temperature, using an upwind scheme at an arbitrary new position. In the present approach, the incompressible Navier–Stokes equations are first solved in a Lagrangian step by a least squares moving particle semi-implicit (LSMPS) method. A redistribution of particle positions is then performed in order to deal with the distorted distribution of particles due to the Lagrangian movement. Finally, the particle velocity is interpolated at the particles’ new position, which is computed by two redistribution models. Taylor–Green vortices, lid-driven flows in a cavity and buoyancy-driven flows in a cavity for several Reynolds and Rayleigh numbers are computed as numerical examples. The results are compared with those of similar approaches proposed previously, such as non-upwind interpolation and the meshless advection using a flow-directional local-grid (MAFL) method. The method presented herein exhibits advantages such as higher accuracy and enhanced numerical stability.

A boundary condition-enforced immersed boundary method for compressible viscous flows

A boundary condition-enforced immersed boundary method (IBM) for simulating compressible viscous flows is presented in this paper. It is developed from the implicit velocity correction-based IBM proposed by Wu and Shu (Journal of Computational Physics 228, 1963–1979 (2009)) for simulation of incompressible viscous flows. In their work, the velocity field is corrected by implementing the no-slip boundary condition. However, their method cannot be directly applied to simulate compressible flows. This is because the immersed boundary has a great effect on velocity, temperature, density and pressure fields in compressible flows, but their method has no mechanism to correct the density and pressure fields. To overcome this difficulty, in this work, the momentum field is firstly corrected by converting the no-slip velocity condition to the momentum condition. After that, from the continuity equation, the density correction can be made from the momentum correction. Then, the corrected velocity field can be given from the corrected momentum field and corrected density. Similar to the correction of momentum field, the temperature field can be corrected from the given temperature condition. Finally, the pressure can be obtained from the equation of state by using the corrected density and temperature. In this way, all the flow variables are corrected and the given physical boundary conditions are accurately implemented. To validate the proposed method, the flows over a stationary cylinder and one or two airfoils at different Mach and Reynolds numbers are simulated. Compared to the results in the literature, good agreement is achieved.

An immersed boundary-gas kinetic flux solver for simulation of incompressible flows

Abstract In this paper, an immersed boundary-gas kinetic flux solver (IB-GKFS) is presented for simulation of incompressible viscous flows. In the present scheme, a simple Cartesian mesh is applied for the flow field and a set of Lagrangian points are used to represent the solid boundary. The solution process of the IB-GKFS can be separated into two steps, the predictor step and the velocity correction step. In the predictor step, the intermediate flow field is obtained by applying the gas kinetic flux solver. As the solid boundary is not considered in this step, there is no external force added in gas distribution function during the evaluation of numerical flux at each cell interface. In the velocity correction step, no-slip boundary condition is imposed at all boundary points to make velocity correction on the surrounding Eulerian points. The implicit boundary condition-enforced immersed boundary method is applied so that no-slip boundary condition can be accurately fulfilled and flow penetration is entirely avoided. The decoupled feature of the predictor step and velocity correction step makes the current scheme simple and efficient because the flux on each cell interface only needs to be calculated once in every time step. With a simple Cartesian mesh and flexible boundary condition treatment, the IB-GKFS can be conveniently applied to solve complex and moving boundary problems. Several numerical experiments are conducted to validate the present scheme, including the flow past a stationary circular cylinder and the NACA0012 airfoil, flow past an in-line oscillating cylinder with prescribed motions. After that, the typical fluid-structure interaction problem of one particle sedimentation in a rectangular domain is further considered. The numerical results of those test cases demonstrate the good capability of the present scheme.

Numerical simulation of two-dimensional viscous flows using combined finite element-immersed boundary method

In this paper, a method that combines the characteristic-based split finite element method (CBS-FEM) and the direct forcing immersed boundary (IB) method is proposed for the simulation of incompressible viscous flows. The structured triangular meshes without regarding the location of the physical boundary of the body is adopted to solve the flow, and the no-slip boundary condition is imposed on the interface. In order to improve the computational efficiency, a grid stretching strategy for the background structured triangular meshes is adopted. The obtained results agree very well with the previous numerical and experimental data. The order of the numerical accuracy is shown to be between 1 and 2. Moreover, the accuracy control by adjusting the number density of the mark points purely at certain stages is explored, and a second power law is obtained. The numerical experiments for the flow around a cylinder behind a backward-facing step show that the location of the cylinder can affect the sizes and the shapes of the corner eddy and the main recirculation region. The proposed method can be applied further to the fluid dynamics with complex geometries, moving boundaries, fluid-structure interactions, etc..