Method For Incompressible Flows Research Articles

A comparative study of split advection algorithms on the moment-of-fluid (MOF) method for incompressible flow

The moment-of-fluid method (MOF) is known as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). In this study, several directional splitting advection algorithms are extended from the VOF-PLIC method to the MOF method. These methods, along with some existing directional splitting algorithms, are presented in detail, especially on the geometrical and non-geometrical nature of the advection algorithm. In addition, a simple and efficient analytic form to calculate the volume and centroid of the intersection polyhedron in 3D is proposed. Several numerical tests, including both 2D and 3D tests, are carried out to investigate mass conservation and geometrical error. Numerical results suggest that the mixed EI and LE scheme (EILE2D) (Aulisa et al., 2003) has the overall best performance in 2D, and Weymouth and Yue’s scheme (WY) has the overall best performance in 3D.

Lattice Boltzmann simulation of the ERCOFTAC pump impeller

The analysis of unsteady flows is becoming more and more important in the field of hydraulic turbomachinery applications, both for stable and especially for unstable operating conditions and other transient phenomena. Unsteady Reynolds-averaged Navier-Stokes (RANS) methods for incompressible flows, implemented via finite volume or finite element approaches, reach their limit in both accuracy and acceptable computational time. The accuracy mainly suffers due to the strong dependence of the solution on turbulence model variants. Furthermore, incompressibility of the flow and the linked implicit pressure equation makes large eddy simulations (LES) of Navier-Stokes-based methods for such flows prohibitively expensive. An alternative to Navier-Stokes fluid solvers are methods based on the solution of the discrete Boltzmann equation, so-called Lattice Boltzmann Methods (LBM). The purpose of this work is to present first results of an ongoing project involving the simulation of unsteady hydraulic turbomachines at high Reynolds numbers. For this purpose, the ERCOFTAC pump impeller was selected as a test case. Results obtained with the current implementation of the in-house code suggest, that the present methodology could provide a viable alternative for the analysis of unsteady flow phenomena in hydromachines.

Efficient SVV stabilized triangular spectral element methods for incompressible flows of high Reynolds numbers

In this paper, we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows. This can be regarded as an extension of a similar stabilization technique for the standard spectral element method. The difficulty of this extension lies in the fact that a suitable definition of spectral vanishing viscosity operator in non-structured elements does not exist, and it is not clear that if a suitably defined spectral vanishing viscosity provides desirable dissipation for the artificially accumulated energy. The main contribution of the paper includes: 1) a well-defined spectral vanishing viscosity operator is proposed for non-standard spectral element methods for the Navier-Stokes equations based on triangular or tetrahedron partitions; 2) an evaluation technique is introduced to efficiently implement the stabilization term without extra computational cost; 3) the accuracy and efficiency of the proposed method is carefully examined through several numerical examples. Our numerical results show that the proposed method not only preserves the exponential convergence, but also produces improved accuracy when applied to the unsteady Navier-Stokes equations having smooth solutions. Especially, the stabilized triangular spectral element method efficiently stabilizes the simulation of high Reynolds incompressible flows.

High order discontinuous Galerkin method for reduced flow models in fractured porous media

We construct a new symmetric interior penalty discontinuous Galerkin (SIPDG)-method for incompressible flow in fractured porous media. The method is developed for computing accurate approximations of the reduced flow model, where fractures are treated as lower dimensional objects. Unlike previous formulations, the methodology proposed herein shows ability to capture the asymptotic limits of highly permeable and fracture seals, also covering a wider range of values of the quadrature integration parameter which appears in the pressure jumps across the fractures. As a first novel contribution we should mention that the method was developed for specific interface condition in the reduced problem for which known in literature DG method are not applicable. As a second novelty we propose a new penalty technique for stabilization of the method and obtain explicit estimate for the penalty parameters associated with flow in matrix and fractures in order to achieve stability. And finally we derive new high order hp type a priori error estimates for the numerical solution in the energy norm. Numerical results illustrate the performance of the proposed SIPDG-method in simulating discrete fracture models.

Improved simplified and highly stable lattice Boltzmann methods for incompressible flows

In this work, improved simplified and highly stable lattice Boltzmann methods (SHSLBMs) are developed for incompressible flows. The SHSLBM is a newly developed scheme within the lattice Boltzmann method (LBM) framework, which utilizes the fractional step technology to resolve the governing equations recovered from lattice Boltzmann equation (LBE) and reconstructs the equations in the Lattice Boltzmann frame. By this treatment, the SHSLBM directly tracks the macroscopic variables in the evolution process rather than the distribution functions of each grid node, which greatly saves virtual memories and simplifies the implementation of physical boundary conditions. However, the Chapman–Enskog expansion analysis reveals that the SHSLBM recover the weakly compressible Navier–Stokes equations with the low Mach number assumption. Therefore, the original SHSLBM can be regarded as an artificial compressible method and may cause some undesired errors. By modifying the evolution equation for the density distribution function, the improved SHSLBMs can eliminate the compressible effects. The incompressible SHSLBMs are compared with the original SHSLBM in terms of accuracy and stability by simulating several two-dimensional steady and unsteady incompressible flow problems, and the results demonstrate that the present SHSLBMs ensure the second order of accuracy and can reduce the compressible effects efficiently, especially for the incompressible flows with large pressure gradients. We then extended the present SHSLBMs to study the more complicated two-dimensional lid-driven flow and found that the present results are in good agreement with available benchmark results.

A mass-momentum consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with capillary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d’Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner “consistent” with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.

A novel median dual finite volume lattice Boltzmann method for incompressible flows on unstructured grids

A novel median dual finite volume lattice Boltzmann method (FV-LBM) for the accurate simulation of incompressible flows on unstructured grids is presented in this paper. The finite volume method is adopted to discretize the discrete velocity Boltzmann equation (DVBE) on median dual control volumes (CVs). In the previous studies on median dual FV-LBMs, the fluxes for each partial face have to be computed separately. In the present second-order scheme, we assume the particle distribution functions (PDFs) to be constant for all faces grouped around a particular edge. The fluxes are then evaluated using the low-diffusion Roe scheme at the midpoint of the edge, and the PDFs at the faces of the CV are obtained through piecewise linear reconstruction of the left and right states. The gradients of the PDFs are computed with the Green–Gauss approach. The presented scheme is validated on four benchmark flows: (a) pressure driven Poiseuille flow; (b) the backward-facing step flow with [Formula: see text], 100, 200 and 300; (c) the lid-driven flow with [Formula: see text] and 1000; and (d) the steady viscous flow past a circular cylinder with [Formula: see text], 20 and 40.

H(div) conforming methods for the rotation form of the incompressible fluid equations

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzman et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the incompressible Navier-Stokes equation with the full version of the stress tensor $$\nu \left( \nabla \varvec{u}+ \nabla \varvec{u}^T - \frac{2}{3} \left( \nabla \cdot \varvec{u}\right) \mathbb {I} \right)$$ , instead of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzman et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order $$p\in \{2,3\}$$ , (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization.

Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.

Block-based adaptive mesh refinement for fluid–structure interactions in incompressible flows

In this study, an immersed boundary (IB) approach on the basis of moving least squares (MLS) interpolation is proposed for analyzing the dynamic response of a rigid body immersed in incompressible flows. An improved mapping strategy is proposed for a quick update of the signed distance field. A CIP-CSL (constraint interpolation profile - semi-Lagrangian) scheme with a compact stencil is adopted for the convective term in momentum equation. Fluid–structure interaction problems can be solved by either the weak or the strong coupling schemes according to the density ratio of the solid and fluid. This research is based on our previous studies on block-structured adaptive mesh refinement (AMR) method for incompressible flows (Liu & Hu, 2018). Present AMR-FSI solver is proved to be accurate and robust in predicting dynamics of VIV (vortex induced vibration) problems. The efficiency of the adaptive method is demonstrated by the 2D simulation of a freely falling plate with the comparison to other numerical methods. Finally, the freely falling and rising 3D sphere are computed and compared with corresponding experimental measurement.

Time consistent fluid structure interaction on collocated grids for incompressible flow

Abstract Consistent time integration on collocated grids for incompressible flow has been studied for static grids using the PISO method, in which the dependencies on time-step size and under-relaxation has been studied in detail. However, for moving grids a detailed description is still missing. Therefore, a step by step analysis of a time consistent fluid–structure interaction (FSI) method for incompressible flow on collocated grids is presented. The method consist of: face normal and area correction for moving grids, treatment of velocity boundary conditions for no-slip walls, time integration of structure equations and fluid force interpolation to structure. The basis of the method is the PISO method of the incompressible Navier–Stokes equations. Time consistency on static grids is shown first, after which time consistency on moving grids is described and analyzed. For moving grids consistent time integration is described in two ways: (1) constructing the face velocities from a normal and tangential component, and (2) correcting the face flux with a face normal and face area correction. For both descriptions the general formulation for the backward differencing schemes (BDF) are given and the correct behavior of the first, second and third order schemes is demonstrated by means of an academic test case (circular cavity flow). Additionally, the (force) coupling from the fluid to the structure is discussed in detail for combining a fourth order explicit Runge Kutta scheme for the structure with a BDF scheme for the fluid. Three interpolations for the fluid forces are shown, which result in either a first order or second order FSI scheme. Third order FSI is demonstrated when the third order BDF scheme is applied on both the structure and fluid equations. Also, under-relaxation for the fluid equations is considered and it is demonstrated that the order of the three BDF schemes are independent of the under-relaxation factor. Finally, the proposed method of time consistent FSI on collocated grids for incompressible flows is demonstrated by applying it to a three-dimensional flow over an elastic structure in a channel and the cylinder flap FSI benchmark case of Turek and Hron.

Development of Moving Particle Explicit (MPE) method for incompressible flows

A new particle method is proposed for computing incompressible flows. Using Moving Particle Semi-implicit (MPS) and Smoothed Particle Hydrodynamics (SPH) approaches, a new Moving Particle Explicit (MPE) method is developed. The concept, like the MPS method, is based on the weighted averaging scheme. Introducing a directional particle number density, the derivation of spatial discretization is performed without imposing any simplification. As a consequence, the formulation becomes more generic and covers the MPS scheme. The solution algorithm is similar to the weakly compressible SPH method. Considering slight compressibility for an incompressible flow, the continuity equation for compressible flow together with an equation of state is used to decouple the velocity and pressure of the Navier–Stokes equations. Consequently, the method becomes fully explicit, and the problems related to solving a set of equations in an implicit procedure are removed. Easy application of the solution algorithm, in comparison with the MPS method, as well as the generic discretization scheme, makes the MPE method more efficient for simulating incompressible flows. The proposed method is validated using a Poiseuille flow and two dam break problems; furthermore, a submerged hydraulic jump is investigated to evaluate the method’s capability in simulating open boundary problems. In all cases, good agreement between numerical results and analytical/experimental data is observed.

A Comparison of Fluid-Cell and Ghost-Cell Direct Forcing Immersed Boundary Method for Incompressible Flows with Heat Transfer

In direct-forcing immersed boundary methods, the forcing terms are generally applied to forcing points on either fluid side (fluid-cell forcing approach) or body side (ghost-cell forcing approach) of the immersed boundary. These direct forcing terms are added to the discretized equations over forcing points in order to implicitly enforce proper boundary conditions on the immersed boundary; hence they dictate the development of the computed flow field. Generally, different forcing approaches adapt different reconstruction models with different stencil supports to compute forcing terms. In this article, a general second-order accurate reconstruction model for solving incompressible Navier-Stokes equations with heat transfer is applied to both the fluid-cell forcing approach and the ghost-cell forcing approach. This allows a meaningful comparison of the solution accuracy and convergence between the two forcing approaches under various boundary conditions. In particular, a simple remedy to reduce the spurious oscillations in pressure force calculation over moving bodies for the ghost-cell forcing approach is devised and verified.

A numerical investigation of explicit pressure-correction projection methods for incompressible flows

A numerical investigation is performed on an explicit pressure-correction projection method. The schemes are fully explicit in time in the framework of the finite difference method. They are tested on benchmark cases of a lid-driven cavity flow, flow past a cylinder and flow over a backward facing step. Comparisons of the numerical simulations have been made with benchmark experimental and DNS data. Based on the results obtained, several numerical issues are discussed; namely, the handling of the pressure term, time discretization and spatial discretization of convective and diffusive terms. The fully explicit projection method is also compared with the fully implicit SIMPLE algorithm. It is observed that the SIMPLE algorithm performs better (faster and produces more accurate results) for laminar flows while the projection method works better for unsteady turbulent flows. Although there have been much research performed using the higher-order pressure incremental projection method, this research work is novel because the schemes employed here are fully explicit, developed in the framework of a finite difference method, and applied to turbulent flows using k-ε model. The major difficulty and challenges of this research work is to identify the sources of instability for the higher-order pressure incremental projection method scheme.

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

AbstractIn this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pairP1—P1which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in theH1-norm for velocity and theL2-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relationh = 𝾪(H2). Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

Transmission–Reflection Coefficient in the Lattice Boltzmann Method

We consider the permeable bounce-back scheme in the lattice Boltzmann (LB) method for incompressible flows, in which a fraction of the distribution function is bounced back and the remainder travels to the neighboring lattice points. An asymptotic analysis of the scheme is carried out in order to show that the fractional coefficient, referred to as the transmission–reflection coefficient, relates the pressure drop to the flow velocity. The derived relation, which clarifies the role played by the transmission–reflection coefficient in the macroscopic description, is helpful in using the scheme to simulate flows involving a pressure drop or gradient. The scheme is compared with the existing methods in which the transmission–reflection coefficient is employed, and the difference is clarified. As an application of the permeable bounce-back scheme, we perform an LB simulation for flows through porous media described by the Brinkman model.

Two-Level Brezzi-Pitkäranta Stabilized Finite Element Methods for the Incompressible Flows

We present a new stabilized finite element method for incompressible flows based on Brezzi-Pitkäranta stabilized method. The stability and error estimates of finite element solutions are derived for classical one-level method. Combining the techniques of two-level discretizations, we propose two-level Stokes/Oseen/Newton iteration methods corresponding to three different linearization methods and show the stability and error estimates of these three methods. We also propose a new Newton correction scheme based on the above two-level iteration methods. Finally, some numerical experiments are given to support the theoretical results and to check the efficiency of these two-level iteration methods.

Modified CIP-CSL/FV method for incompressible flows

This paper presents a modified staggered-like CIP-CSL/FV (constrained interpolation profile-conservative semi-Lagrangian/finite volume) method for incompressible flows. In the present method, which is originated in VSIAM3 (volume/surface integrated average based multi-moment method), the surface-integrated values are employed as the major variables to increase the connectivity between the volume-integrated and surface-integrated values and between the collocated volume-integrated values. Resultingly, it changes the solution procedure by modifying the trajectory averaging process of all moments having been introduced in the original researches. To verify the staggered-like CIP-CSL/FV method, two phase benchmarking problems are tested with the THINC/WLIC (tangent of hyperbola for interface capturing/weighted line interface calculation) scheme capturing interfaces sharply and volume-conservatively.

SPACE–TIME VMS METHODS FOR MODELING OF INCOMPRESSIBLE FLOWS AT HIGH REYNOLDS NUMBERS

Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation was developed for flow problems with moving interfaces and has been successfully applied to some of the most complex problems in that category. A new version of the DSD/SST method for incompressible flows, which has additional subgrid-scale representation features, is the space–time version of the residual-based variational multiscale (VMS) method. This new version, called DSD/SST-VMST and also Space–Time VMS (ST-VMS), provides a more comprehensive framework for the VMS method. We describe the ST-VMS method, including the embedded stabilization parameters, and assess its performance in computation of flow problems at high Reynolds numbers by comparing the results to experimental data. The computations, which include those with 3D airfoil geometries and spacecraft configurations, signal a promising future for the ST-VMS method.

A subgrid computation of the curvature by a particle/level-set method. Application to a front-tracking/ghost-fluid method for incompressible flows

A particle/level-set method is developed to capture the interface between two immiscible phases. No redistance equation is used for the level-set function which is built from an accurate cubic spline interpolation of the markers describing the interface. Mass losses which usually appear with level-set functions are drastically reduced. Interface coupling with the incompressible momentum equation is taken into account thanks to the ghost-fluid method. Indeed, the particles allow an accurate computation of the local interfacial curvatures, and the capillary part of the pressure jump computed across the interface is more accurately evaluated. Results on static drop show a large decrease of spurious currents compared to other methods. Scalar advection validations and convergence studies are carried out as well as tough test-cases involving large density ratios (typically air/water). A better agreement with literature results is shown compared to classical approaches.