Incompressible Flow Problems Research Articles

A mimetic method for polygons

A new method for mimetic interpolation on polygonal meshes is described. This new method is based on harmonic function interpolation. Explicit formulas for harmonic functions on general polygons do not exist, so truncated harmonic polynomial expansions are used for computational efficiency. We show that the naive harmonic polynomial expansion, is not stable for arbitrary polygons. However, higher level truncations of harmonic interpolations are stable and accurate. This new method is shown to be a direct extension of the lowest order Raviart-Thomas finite elements to polygons. This method is also a direct extension of the finite volume MAC method to polygons. The accuracy of the interpolation is shown to be first-order irrespective of the polynomial truncation level. However the accuracy of vector Laplace equation solutions using this inner product is shown to be second-order accurate, in keeping with other lowest order Mimetic methods. The versatility of this numerical method is demonstrated on a multiphase incompressible flow problem with a density jump of 1000, on a moving polygonal mesh.

High-order upwind compact finite-difference lattice Boltzmann method for viscous incompressible flows

In this work, a high-order upwind compact finite-difference lattice Boltzmann method (UCDLBM) is developed to efficiently solve viscous incompressible flow problems. A fifth-order upwind compact difference scheme is adopted to discretize the spatial derivatives of the lattice Boltzmann equation, and the third-order total-variation-diminishing Runge–Kutta scheme is utilized for the discretization of the temporal term. Compared to the existing central compact finite-difference lattice Boltzmann method (CFDLBM), the present UCDLBM can prevent non-physical oscillations without filtering due to the natural dissipative property of upwind schemes. Three benchmark problems involving the Taylor–Green vortex problem, the doubly periodic shear layer flow problem and the lid driven square cavity flow problem are numerically solved to demonstrate the accuracy and efficiency of the present method. Numerical results computed are in good agreement with the analytical solution or other available numerical results. And, the present UCDLBM is less time-consuming than the CFDLBM without degenerating the order of accuracy of the numerical solutions.

A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier–Stokes equations

The present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element-free Galerkin (IEFG) method, for the numerical solution of incompressible fluid flow problems. In principle, the numerical procedures to be implemented in this communication have been conceived for finite element method (FEM)-based solutions, and these include the reduced integration penalty method (RIPM), the streamline upwind Petrov–Galerkin (SUPG) scheme, and a penalty-based immersed boundary method (PBIBM) for the imposition of essential boundary conditions along internal fluid–solid interfaces. The linear momentum balance and mass conservation equations have been coupled via the RIPM, in order to obtain a global weak formulation where the IEFG model is entirely developed in terms of improved moving least squares (IMLS) approximations of the velocity field. A detailed explanation concerning the appropriate extension of both the RIPM and SUPG procedures to the context of IEFG formulations, has also been provided. The resulting formulation has been applied to the solution of two well-known benchmark problems: i) Lid-driven square cavity flow, and ii) Flow past a fixed cylinder. Regarding the flow past a fixed cylinder benchmark problem, the fluid–solid interaction has been imposed as an internal immersed boundary condition via the PBIBM. The feasibility and reliability of implementing the RIPM, SUPG and PBIBM procedures in the IEFG formulation, have been proven by comparison with experimental and mesh-based numerical results reported in the literature. The results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the IEFG formulation, allows the achievement of accurate and stable numerical results during the solution of incompressible fluid-dynamics problems.

A residual-based stabilized finite element formulation for incompressible flow problems in the Arlequin framework

Many fluid flow problems involve localized effects within a larger flow domain, with boundary layers close to solid boundaries being one of the most common. Accurate and realistic computational analysis of such problems requires that the local flow behavior be properly represented by a numerical formulation with affordable computational cost. Among several strategies developed in the computational mechanics to deal with multiscale phenomena, the Arlequin method proposes overlapping a local discretization to a global one, in the neighborhood of localized effects, and gluing both models in an appropriate subzone of the overlap by means of Lagrange multipliers method, being successful in the solid mechanics context, but less explored in CFD analysis. To improve stability and conditioning of the algebraic system of equations and, at the same time ensure more flexibility, we propose a novel residual-based stabilized Arlequin formulation. Following, the proposed formulation is applied to the numerical analysis of incompressible flows. The resulting formulation is tested with selected numerical examples, considering structured and unstructured, coincident and non-coincident finite element discretization, Stokes problems and convection dominated Navier–Stokes problems, steady and transient cases, showing the methodology precision, robustness and flexibility.

A multinumerics scheme for incompressible two-phase flow

The incompressible two-phase flow problem is solved by a method that combines cell-centered finite volume with discontinuous Galerkin in non-overlapping subdomains. The primary unknowns are the wetting phase pressure and the capillary pressure. The nonlinear equations are solved fully implicitly at each time step. Fluxes at the interface between subdomains are defined implicitly to allow for seamless propagation of saturation fronts. Numerical results show the robustness and efficiency of the method for homogeneous and heterogeneous porous media.

Incompressible Viscous Flow Simulation Using the Quasi-Hydrodynamic Equations’ System

In this paper, we consider the problem of modeling viscous incompressible fluid flow using the quasi-hydrodynamic (QHD) system of equations. The use of this approach makes it possible to avoid instability in the pressure calculation when using the classical formulation of the Navier–Stokes equations, which is manifested when using cellular numerical schemes. For the numerical implementation of QHD equations, the finite volume method is used with cell-center approximation. In the case of a two-dimensional problem a square mesh is used. A cubic mesh is used in the case of a three-dimensional problem. Two series of calculations are performed for testing in areas of complex geometry for several Reynolds numbers. The results are compared with the ANSYS CFX software package. The comparison shows the high quality of the numerical simulation results using the QHD system of equations.

Numerical Simulation of Laminar Taylor–Couette Flow

The problem of the viscous incompressible fluid flow between two coaxial cylinders is considered in the case when the inner cylinder is rotates and the outer cylinder is stationary. The process of fluid flow is simulated by solving the three-dimensional Navier–Stokes equations. The computational algorithm is based on the finite volume method. The dependences of the dimensionless viscous torque from the Reynolds number and the radius ratio are established.

Improving accuracy of the moving grid particle finite element method via a scheme based on Strang splitting

Particle finite element method (PFEM) is a computational tool suitable for simulating fluid dynamics problems characterized by presence of moving boundaries. In this paper a new version of the method for incompressible flow problems is proposed aiming at accuracy improvement. This goal is achieved essentially by applying Strang operator splitting to Navier–Stokes equations and selecting adequate integration schemes for the resulting advective and Stokes sub-problems. For achieving efficient implementation, the pressure and the velocity in the Stokes part are decoupled via the fractional step technique as in the classical PFEM. However, at the first fractional step an explicit pressure prediction procedure for alleviating mass losses is introduced. Three test cases are solved, validating the methodology and estimating its accuracy. The numerical evidence proves that the proposed scheme improves the accuracy of the PFEM.

An iterative divergence-free immersed boundary method in the finite element framework for moving bodies

A novel iterative direct-forcing immersed boundary (IB) method, which meets the divergence-free and no-slip conditions simultaneously, is proposed in the finite element framework for incompressible flow problems. The coupled velocity and the pressure of the fluid field are solved by the Characteristic-based Split scheme. The formulation for the extra body force is derived according to the no-slip boundary condition. The novelty of the proposed IB method lies in that the calculation of the pressure, the iterations of the velocity, the pressure and the extra body force are accomplished in a loop at the same time. Therefore, the divergence-free and no-slip conditions are ensured fully at the same time. The accuracy of the current IB method is verified by several cases, including static and moving boundaries. Meanwhile, the contributing factors of the spurious force oscillations for moving boundaries are studied. A brand-new viewpoint is proposed, which is different with the common standpoint. The major source of spurious force oscillations pointed out in this paper is that the change of interpolation coefficients when boundaries sweep within an element, instead of the change of interpolation points when boundaries move across adjacent elements, as discussed in existing references. The oscillations which are caused by the inherent properties of the immersed boundary method cannot be avoided, while effective methods can be employed to suppress them, including refining the meshes and applying a proper discrete delta function.

Numerical verification of a non-residual orthogonal term-by-term stabilized finite element formulation for incompressible convective flow problems

The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual based formulations, the introduction of two velocity subscale components. The theoretical rate of convergence of the method is verified numerically using linear and quadratic equal-order finite element discretizations. To this end, a standard convergence test of L2-norm is presented where the computed solutions are compared when manufactured solutions are imposed at Gauss-point level. Moreover, the Hopf bifurcation is studied for two well-known benchmark problems: flow past a cylinder and the three-dimensional lid-driven cavity flow. For the flow past a cylinder case, the Hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution. In particular, a dominant convective problem (Re=4000) is solved using both the quasistatic and dynamic versions of the method, evaluating the performance of each one in the quality of the solution and in the CPU time needed to obtain a converged solution. For the 3D lid-driven cavity flow problem, the Hopf bifurcation is determined using two different boundary conditions, analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer. The final test case is the turbulent 3D lid-driven cavity problem (Re=12000), where velocity profiles are compared with experimental, LES and DNS reference solutions. Additionally, pressure and velocity spectra are shown at certain representative points of the domain, as well as phase diagrams, correlation function graphs, the Poincaré map, and the Lyapunov exponent, typical mathematical tools used in dynamical system analysis. From the results, the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones.

Evaluation of a nonlinear variational multiscale method for fluid transport problems

Diverse transport problems, especially those based on fluid flow models, are intrinsically multiscale and nonlinear, characteristics that often lead to intricate dynamics such as the development of instabilities and turbulence. Computational simulations that resolve all scales in these problems are often unfeasible, prompting to coarse-grained simulation strategies in which small-scale features are modeled instead of resolved. Variational Multiscale (VMS) methods, and particularly residual-based Large-Eddy Simulation (LES) approaches, have proven effective and robust for the coarse-grained simulation of complex transport problems. VMS methods avoid the assumption of separable nonlinearity and the reliance on empirical small-scale models by using a variational decomposition of scales together with a residual-based approximation of the small-scales. Evaluation of a nonlinear VMS approach, denoted as VMSn, is presented for the coarse-grained simulation of transient-advective-diffusive-reactive (TADR) transport problems arising from fluid flow models. In contrast to classical VMS approaches that neglect the effect of the small scales on the transport operator, VMSn treats the inter-dependence between large- and small-scales upfront. The treatment of inter-scale coupling involves the solution of a local algebraic nonlinear system describing the evolution of the small-scales. The VMSn approach is complemented with two algebraic approximations of the small-scales: one based on the main diagonal of the transport matrices and another that preserves transport fluxes and is suitable for generic TADR systems. The suitability of the VMSn approach for handling general TADR problems and regimes is evaluated with benchmark incompressible, compressible, and magnetohydrodynamic laminar flow problems, the incompressible Taylor-Green vortex flow, the turbulent free jet, and the two-temperature arc in crossflow. Simulation results show that VMSn leads to minor improvements in accuracy with respect to the classical VMS for the laminar flow problems, but to significantly greater accuracy for the turbulent flows and the unsteady plasma flow problems, while using the same cohesive numerical formulation.

A reconstructed discontinuous Galerkin method for incompressible flows on arbitrary grids

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific, the inviscid term of the INS equations is discretized by applying the third-order hybrid rDG(P1P2) method with a simplified artificial compressibility flux, while the viscous term of the INS equations is discretized by using the simple direct DG (DDG) method. A number of incompressible flow problems, in both steady and unsteady forms, for a variety flow conditions are computed to numerically assess the performance of the hybrid rDG(P1P2) method, which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs. Furthermore, a detailed comparison of a variety of different reconstructed strategies is performed and presented. Numerical results demonstrate that the hybrid rDG(P1P2) method outperforms the rDG(P1P2) method based on either the original least-squares reconstruction or the Green-Gauss reconstruction for solving the INS equations.

Hp-discontinuous Galerkin Finite Element Method for Incompressible Miscible Displacement in Porous Media

We used (hp-DGFEM) for solving the coupled system of incompressible miscible displacement problem in porous media. Flow and transport equations are solved by hp-symmetric interior penalty Discontinuous Galerkin (hp-SIPG) method. We prove the error analysis in the energy norm to the coupled system of incompressible one-phase flow problem.

A parallel inverted dual time stepping method for unsteady incompressible fluid flow and heat transfer problems

The simulation of unsteady incompressible fluid flow and heat transfer problems is a huge time-consuming task and has been becoming one of the hottest topics in the fields of computational fluid dynamics and numerical heat transfer. Pure spatial parallelization has been so far the most widely used parallel strategy to speed up unsteady simulations but suffers from the problem of speedup saturation as parallel scale increases. In order to further speed up unsteady simulations and exploit the concurrency in temporal dimension, we propose a novel temporal parallelization strategy, i.e., parallel inverted dual time stepping method, by simply inverting the sequence of the inner loop (pseudo time) and the outer loop (physical time) within the dual time stepping framework. The parallel performance of the proposed method is verified and validated with three steady/unsteady problems. Parallel results of the proposed method agree well with those of the dual time stepping method and the available benchmark solutions. The parallel efficiency of the proposed method is about 96%, 81%, 62%, 51%, 45% for 4, 8, 12, 16, 20 CPU cores, respectively, which is considerably higher than that of the widely used parareal method. The proposed method has potential in further increasing parallel scale as spatial parallelization is saturated.

A single domain velocity - vorticity Fast Multipole Boundary Domain Element Method for three dimensional incompressible fluid flow problems, part II

In the present work, the Fast Multipole Boundary Element Method (FMM / BEM) for solving three-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations is proposed. With the velocity-vorticity formulation the pressure gradient is eliminated from the equations. The kinematics equation, related to the velocity field satisfies continuity and provides a direct boundary condition for the vorticity equation. The single-domain approach is used for the discretization of the entire computational volume. The system of equations is compressed into two vectors and a preconditioner matrix, which is negligible in size. The involved unknowns are velocities, vorticities, and boundary vorticity fluxes. The two governing equations are coupled together in a convergent Newton–Raphson iteration scheme, successfully used for the solution of 3D fluid flow problems on a 32 GB memory computer. The degrees of freedom of the benchmark problems are above to 300000, which is an unreachable limit for the conventional single-domain BEM.

A high-order discontinuous Galerkin method for simulating incompressible fluid-thermal-structural interaction problems

This paper presents a framework for the application of the discontinuous Galerkin(DG) finite element method to the multi-physics simulation of the solid thermal deformation interacting with incompressible flow problems in two-dimensions. Recent applications of the DG method are primarily for thermoelastic problems in a solid domain or fluid-structure interaction problems without heat transfer. Based on a recently published conjugate heat transfer solver, the incompressible Navier-Stokes equation, the fluid advection-diffusion equation, the Boussinesq term, the solid heat equation and the solid linear elastic equation are solved using an explicit DG formulation. A Dirichlet-Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux computed at interface quadrature points in the fluid-solid interface. Formal hp convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved for each solver. The algorithm is then further validated against several existing benchmark cases including the in-plane loaded square, the Timoshenko Beam, the laminated beam subject to thermal-loads and the lid-driven cavity with a flexible bottom wall. The computational effort demonstrates that for all cases the highest order accurate algorithm has several magnitudes lower error than the second-order schemes for a given computational effort. It is a strong justification for the development of such high order discretisations. The solver can be employed to predict thermal deformation of a structure due to convective and conductive heat transfer at low Mach, such as chip deformation on a printed circuit board, wave-guide structure optimization, thermoelectric cooler simulation, and optics mounting method verification.

Numerical study of the motion of a single elongated bubble in high viscosity stagnant liquids along pipelines

Multiphase flow models involve more variables than available equations, thus closure relationships are required to solve the system of partial differential equations. In the Drift Flux Model, the mean drift velocity of the gas phase is estimate using the drift velocity closure relationship. This study investigates the motion of a single elongated bubble in high viscosity stagnant liquids in order to propose a comprehensive closure relationship for the bubble drift velocity. Different cases were run with inclination angles varying from 0° (horizontal direction) to 90° (vertical upward). OpenFOAM, an open source Computation Fluid Dynamics software, was used to numerically solve the two-phase incompressible flow problem. Drift velocities based on the movement of the bubble tip and the void fraction weighted averaged were compared to clarify the ambiguity in the drift velocity calculation and measurement. For horizontal pipe cases, the deformation of the bubble caused the drift velocity to be higher at the tip of the bubble than the drift velocity inside the bubble. For inclination angles more than 0°, the drift velocity at the tip is the same as inside the bubble. The closure relationship proposed is validated against experimental data in the literature. The results indicate that the closure relationship has a maximum 10% absolute average relative error for the viscosity range of 0.14–1.120 Pa·s, pipe size up to 0.152 m and pipe inclinations from 0° to 90°. The limitations of the closure relationship and potential enhancements for future studies are discussed.

Parabolic evolution equations in interpolation spaces: boundedness, stability, and applications

We develop a general approach for the study of the boundedness and stability of solution to general parabolic semi-linear evolution equations in real-interpolation spaces. Our approach yields a unified method to handle problems in Newtonian viscous incompressible fluid flows. Moreover, our method will be applied to derive new results for the existence and polynomial stability of bounded solutions to the Ornstein–Uhlenbeck semi-linear equations and various diffusion equations with rough coefficients. The method is based on a combination of interpolation functors, duality estimates, smoothing properties of the linearized equation, and fixed point arguments.

Adaptive discontinuous Galerkin methods for solving an incompressible Stokes flow problem with slip boundary condition of frictional type

We consider the numerical solution of a variational inequality arising in the study of an incompressible Stokes flow with a nonlinear slip boundary condition of frictional type. We study several discontinuous Galerkin (DG) schemes for solving the problem, and derive reliable and efficient residual type a posteriori error estimators. An adaptive DG method is constructed based on the error estimators to solve the problem. Some numerical examples are presented to illustrate the efficiency of the adaptive DG method.

Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems

SummaryMonolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two‐level generalized Dryja‐Smith‐Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.