Momentum Interpolation Method Research Articles

On wave-current interaction with a horizontal cylinder near free surface by a fourth-order-accurate finite volume compact solver

We present a numerical solver for dealing with the problem of wave-current interaction with a horizontal cylinder near free surface. The solver solves the filtered Navier-Stokes equations which are discretized using a fourth-order-accurate compact finite volume scheme on a collocated grid. The turbulence effect is modelled by a subgrid-scale model using a large eddy simulation, and the free surface and the cylinder are modelled by a volume of fluid method and a ghost cell immersed boundary method. The pressure-velocity coupling is ensured via a momentum interpolation method. We obtain the hydrodynamic force and its spectra characteristics, dynamic pressure distribution, streamline topology and vorticity patterns with different immersed depths under uniform currents, regular waves and wave-current flows. Validation is performed by comparing the calculated hydrodynamic force with the measured data within one wave period. The results show that the position of vortex shedding and dissipation is different for the cases under different immersed depth in regular waves and wave-current flows. The dynamic pressure around the cylinder as the force reaches its peak value changes with immersed depth. The value and phase of the force are sensitive to immersed depth.

On ghost fluid method-based sharp interface level set method on a co-located grid and its comparison with balanced force-based diffuse interface method

This paper, on Computational multi-Fluid Dynamics (CmFD) development, presents an extension of the Ghost Fluid Method (GFM)-based Sharp Interface Level Set Method (SI-LSM) (originally proposed on a staggered grid) for a co-located grid system. Further, this paper presents a comparative study for the GFM-based SI-LSM and a balanced force method (BFM)-based Diffuse Interface Level Set Method (DI-LSM). The BFM-based DI-LSM avoids a pressure-interfacial force decoupling, during the implementation of the MIM, by a proper-discretization of the surface tension term; not needed for the single-phase flow. Whereas, the GFM-based SI-LSM implicitly couples two immiscible, incompressible fluids via interface jump condition, which makes the Momentum Interpolation Method (MIM) for the two-fluid flow same as that for a single-fluid flow. A unified, for both SI-LSM and DI-LSM, mathematical and numerical formulations are presented. Order of accuracy is demonstrated in-between first-order and second-order; similar to that reported earlier for the various CmFD methods. Finally, validation and relative-performance study are presented for four test cases: static droplet, dam break, drop coalescence, and Rayleigh-Taylor instability. For the static droplet, the relative-performance of the present SI-LSM as compared to the DI-LSM is found similar to that in the literature for SI-VOF (Volume of Fluid) method as compared to DI-VOF method. For the various test cases, superior performance is demonstrated for the SI-LSM as compared to DI-LSM. This paper presents a broader perspective to the MIM for a single-versus-two fluid-flow and diffuse-versus-sharp approaches; and a novel GFM-based SI-LSM on a co-located grid.

Multi-point momentum interpolation correction on collocated meshes

Since the momentum interpolation method was proposed by Rhie and Chow in 1983, it has been widely used in studies of Computational Fluids Dynamics (CFD). The conventional momentum interpolation methods were designed across the edge between two neighboring cells. In this study, an alternative momentum interpolation method, called multi-point momentum interpolation correction (IC) method, is proposed. The proposed IC method is distinguished from the conventional cross-edge momentum interpolation methods by correcting and improving the edge velocity with the interpolated values of its surrounding edges. Examples including analytic, experimental and field cases demonstrated that the proposed IC method is generally capable of improving the convergence process and numerical accuracy.

A ghost-cell immersed boundary method on preventing spurious oscillations for incompressible flows with a momentum interpolation method

We propose a ghost-cell immersed boundary method (GCIBM) using a local directional ghost cell approach for the simulation of incompressible flows involving solid geometry. The proposed method is based on a finite volume solver with a momentum interpolation method (MIM) on a collocated grid for the Navier-Stokes equations that is spatially discretized by a fourth-order compact scheme and temporally advanced by a fourth-order Runge-Kutta scheme. A new stencil for imposing the Neumann boundary is constructed following the least square regression. The spurious oscillations that spread from the immersed boundary are observed when the discrete momentum forcing (DMF) is obtained at the cell centers for velocities. This is due to the decoupling of the pressure and velocity fields. In this paper, the spurious oscillations are removed by the recovery of the decoupling by shifting the locations of DMF from the cell centers to the face centers. The efficiency of the proposed technique is confirmed in terms of a simulation for steady and unsteady flows with a stationary or moving geometry.

A free element scheme for simulating two‐ and three‐dimensional incompressible fluid flow

AbstractIn this work, Free Element method (FECM) is extended to solve incompressible Navier‐Stokes equations. The momentum equations are discretized by FECM, which permits overlapped elements. Then, the velocities at midpoints are interpolated by improved Momentum Interpolation Method to avoid oscillation caused by decoupling of velocity and pressure. At last, the pressure correction equation is solved to make sure the results satisfying continuity equation. The interpolation of midpoint's velocity is proved to be independent of the under‐relaxation factor and time step size. The deferred correction is employed for utilization of higher‐order discretization of convective terms. Taylor‐Green vortex, flow over a cylinder, lid‐driven flow, cooling tunnel, and flow over airfoil are simulated. The results computed by the proposed method are compared with exact solutions and benchmark solutions, which indicates that the new method has the second order accuracy in space. What is more, the proposed method works well on random node distributions as well, which means that the method is much robust.

Basic verification of a numerical framework applied to a morphology adaptive multifield two‐fluid model considering bubble motions

SummaryA morphology adaptive modeling framework is derived that is able to handle computationally efficiently dispersed as well as resolved interfacial structures coexisting in the computational domain with the same set of equations. The Eulerian multifield two‐fluid model is combined with the compact momentum interpolation method for multiple phases, which has been proposed in the literature as an extension to the Rhie‐Chow pressure‐velocity coupling. Additionally to the interfacial drag force, the virtual mass force is consistently accounted for in the model. Utilizing a specialized interfacial drag formulation, large interfacial structures can be described with the presented method in a volume‐of‐fluid‐like manner, additionally to the disperse description. The strong phase coupling due to the drag closure model in interfacial regions is resolved with a partial elimination algorithm, which is adapted to work in an approximate manner for more than two phases via a sum formulation. The presented model is implemented in the C++ library OpenFOAM and solver performance is compared with results obtained with the homogeneous model approach in two cases of a single rising gas bubble for two‐ and three‐dimensional space, respectively. Additionally, for both three‐dimensional cases, the results are compared with experimental data. Finally, the presented method's capability of representing dispersed and resolved interfacial structures at the same time is demonstrated with two test cases: a two‐dimensional gas bubble, rising in a liquid, which is laden with micro gas bubbles, and a two‐dimensional stagnant stratification of water and oil, sharing a large‐scale interface, which is penetrated by micro gas bubbles.

An unstructured finite volume method based on the projection method combined momentum interpolation with a central scheme for three-dimensional nonhydrostatic turbulent flows

This paper presents a three-dimensional nonhydrostatic model to solve the Navier–Stokes equations using an unstructured finite volume method. The physical domain could be geometrically arbitrary. To avoid the checkerboard problem caused by non-staggered grids, a momentum interpolation method is used by introducing face-normal velocities at the mid-points of the cell faces. As the Large Eddy Simulation (LES) requires at least second-order accuracy in time and in space for all the terms, a central scheme combined with an explicit Adams–Bashforth scheme is proposed in this model. The projection method is applied to decouple the velocity field and pressure. Several benchmark test cases are used to validate the second-order accuracy, the numerical stability and the performance of the model. Analysis on divergence noise using an unstructured collocated triangular grid, as well as on the ratio between vertical and horizontal spacing steps have been done to show the reliability of the model. The proposed model has been used to simulate backward-facing step flows, lid-cavity flows, turbulent open channel flows and the turbulent flows around a vertical cylinder. The convergence of the linear solver is analyzed in terms of the iterations and CPU time. The results are fairly in agreement with the references in the literature. The proposed model is able to correctly reproduce the characteristic flow features in all the test cases.

A parallel second-order unstructured finite volume method for 3D free-surface flows using a σ coordinate

In this paper, we introduce a second-order time- and space-accurate technique, developed to solve in parallel free-surface flows in arbitrary three-dimensional geometries. The discretization is based on a second-order finite-volume technique on prisms elements, consisting of triangular grids on the horizontal and bounded by a free surface and an irregular bottom on the vertical. The equations are transformed vertically to the σ-coordinate system in order to obtain an accurate representation of top and bottom topography. The reconstruction of three presure/velocity decoupling methods using a Crank-Nicolson scheme formulation is proposed. The Momentum Interpolation Method (MIM) is combined with Local Extremum Diminishing (LED) second-order upstream scheme for convective terms is developed. The parallelization is designed by a block domain decomposition technique. The discretization results in non-symmetric variable-coefficient linear systems which are solved using a parallel multi-color Successive Over-Relaxation algorithm. Several test cases of surface wave motion are used to demonstrate the capabilities, numerical stability and performance of the model.

Prediction of roll hydrodynamic characteristics for a ship hull section with a high-order finite volume solver

This work predicts hydrodynamic characteristics for a rolling ship hull section. Flows about the mid-section of a KVLCC2 under harmonic excited roll motions are simulated with solutions of Navier-Stokes (NS) equations filtered by a large eddy simulation (LES) with a finite volume (FV) solver. This solver applies a fourth-order compact scheme to discrete spatial terms, a fourth-order Runge-Kutta scheme to advance equations, a momentum interpolation method (MIM) to avoid pressure and velocity decoupling, a volume of fluid (VOF) method to capture free surface, and a ghost cell immersed boundary method (GCIBM) to simulate rolling ship. Firstly, verification is implemented to confirm the convergency. Validation is followed by comparison with experimental data. Then, systematic assessments are made on the effects of roll amplitude, frequency and bilge keels by analyzing the tendency of roll moment, damping and added mass moment of inertia as well as flow structures. Wave radiation and viscous effects are elucidated through flow structures. As a result, viscous effect is dominant. Roll amplitude, frequency and bilge keels influence roll hydrodynamic characteristics substantially. Besides, causes of vortex generation, shedding and movement, variation of dynamic pressure field and mechanism of bilge keel to mitigate roll motion are disclosed.

Implementation of an implicit pressure–velocity coupling for the Eulerian multi-fluid model

An implicitly coupled pressure–velocity transient scheme for the solution of the Eulerian multi-fluid model with any number of phases (MIC) was developed and implemented in OpenFOAM®. The numerical methodology is based on the phase-intensive momentum equations for the phase velocities and a pressure equation using a deferred-correction approach of the Rhie–Chow interpolation. An extension of the compact momentum interpolation method (CMI) to a system with any number of phases was developed. The developed methodology was tested against a segregated multiphase solver (MS) and a steady-state single phase solver, in cases considering up to four phases. The proposed method has proven to be more robust and accurate than the segregated counterpart, being able to converge in cases with large drag coefficients, providing time-step independent steady-state solutions.

Modifications to the SIMPLE algorithm with the MDCD approach for incompressible flow simulation

PurposeThe purpose of the paper is to propose some modifications to the SIMPLE (semi-implicit method for pressure-linked equations) algorithm. These modifications can ensure the numerical robustness and optimize computational efficiency. They remarkably promote the ability of the SIMPLE algorithm for incompressible DNS (direct numerical simulation) of multiscale problems, such as transitional flows and turbulent flows, by improving the properties of dispersion and dissipation.Design/methodology/approachThe MDCD (minimized dispersion and controllable dissipation) scheme and MMIM (modified momentum interpolation method) are introduced. Six typical test cases are used to validate the modified algorithm, including the linear convective flow, lid-driven cavity flow, laminar boundary layer, Taylor vortex and DHIT (decaying homogenous isotropic turbulence). Particularly, a highly unsteady DNS of separated-flow transition in turbomachinery is precisely predicted by the modified algorithm.FindingsThe numerical examples show the distinct superiority of the modified algorithm in both internal flows and external flows. The advantages of the MDCD scheme and MMIM make the SIMPLE algorithm a promising method for DNS.Originality/valueSome effective modifications to the SIMPLE algorithm are addressed. It is the first attempt to introduce the MDCD approach into the SIMPLE-type algorithms. The new algorithm is especially suitable for the incompressible DNS of convection-dominated flows.

Role of the momentum interpolation mechanism of the Roe scheme in shock instability

SummaryThe shock instability phenomenon is a well‐known problem for hypersonic flow computation by the shock‐capturing Roe scheme. The pressure checkerboard is another well‐known problem for low‐Mach‐number flow computation. The momentum interpolation method (MIM) is necessary for low‐Mach‐number flows to suppress the pressure checkerboard problem, and the pressure‐difference‐driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high‐Mach‐number flows or low‐Mach‐number cells in numerical shock. Therefore, MIM should be retained for low‐Mach‐number flows and be completely removed for high‐Mach‐number flows and low‐Mach‐number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.

A pressure-based Mach-uniform method for viscous fluid flows

ABSTRACTA pressure-based, Mach-uniform method is developed by combining the SLAU2 numerical scheme and the higher temporal order pressure-based algorithm. This hybrid combination compensates the limitation of the SLAU2 numerical scheme in the low-Mach number regime and deficiencies of the pressure-based method in the high-Mach number regime. A momentum interpolation method is proposed to replace the Rhie-Chow interpolation for accurate shock-capturing and to alleviate the carbuncle phenomena. The momentum interpolation method is consistent in addition to preserving pressure–velocity coupling in the incompressible limit . The postulated pressure equation allows the algorithm to compute the subsonic flows without empirical scaling of numerical dissipation at low-Mach number computation. Several test cases involving a broad range of Mach number regimes are presented. The numerical results demonstrate that the present algorithm is remarkable for the calculation of viscous fluid flows at arbitrary Mach number including shock wave/laminar boundary layer interaction and aerodynamics heating problem.

Influence of momentum interpolation methods on the accuracy and convergence of pressure–velocity coupling algorithms in OpenFOAM®

Checkerboard pressure is one of the most significant problems arising from the use of collocated grids for fluid dynamic simulations using the finite-volume method. The original Rhie–Chow momentum interpolation technique, termed Original Momentum Interpolation Method (OMIM) was proposed to eliminate the non-physical saw-tooth pressure oscillations. However, it was soon proved that the steady-state solutions obtained with this technique were under-relaxation factor dependent. Nevertheless, standard OMIM is still commonly used in several CFD codes like OpenFOAM®, the software used for this work. In this paper the OMIM and a possible correction for under-relaxation dependency, which has been implemented in OpenFOAM®, are discussed in detail. The proposed methodology is compared and contrasted with OMIM in terms of accuracy of the solution and speed of convergence for several classical pressure–velocity segregated algorithms for steady state solvers; namely SIMPLE, SIMPLE-C, SIMPLE-R and PISO. A classical laminar 2D cavity is used as the base test-case. The study is then extended to a more complex 2D airfoil profile (NACA0012). In the cases considered mesh uniformity and orthogonality are progressively reduced and turbulence starts playing an important role, limiting therefore the convergence of the cases and the performance of the correction.

A parallel VOF IB pressure-correction method for simulation of multiphase flows

This paper develops an improved pressure correction method, coupled with volume-of-fluid (VOF) and immersed boundary (IB) methods, to accelerate multiphase flow computations. A momentum interpolation method (MIM) is used to avoid the issue of a checkerboard pressure field on a non-staggered grid system. A transpose-free quasi-minimal residual (TFQMR) method is applied to reduce the CPU time when solving the Poisson equation for the pressure correction procedure. A suitable initial guess for the TFQMR method is investigated to enhance the stability of the matrix solver. The IB method is applied to treat the solid-wall boundary conditions while OpenMP is used to reduce CPU time. Two physical problems – dam breakage and a water wave impacting on a tall structure – are simulated and compared with known data. The improved numerical scheme has CPU speeds at least 8 times faster than the initial numerical scheme for simulating two-phase flows. With regard to the parallel computational efficiency, using six cores with the OpenMP allows the improved scheme to have 77% parallel efficiency. These modifications greatly enhance overall computational accuracy and efficiency.

Solving low Mach number Riemann problems by a momentum interpolation method

A momentum interpolation based scheme is proposed, giving satisfactory acoustic solutions in low Mach number regime.

A new higher-order finite volume method based on Moving Least Squares for the resolution of the incompressible Navier–Stokes equations on unstructured grids

In this work a new higher-order (>2) accurate finite volume method for the resolution of the incompressible Navier–Stokes equations on unstructured grids is presented. The formulation is based on the use of Moving Least Squares (MLS) approximants. Third and fourth order accurate discretizations of the convective and viscous fluxes are obtained on collocated meshes. In addition, MLS is employed to design a new Momentum Interpolation Method that allows interpolations better than linear on any kind of mesh. The accuracy and performance of the proposed method is demonstrated by solving different steady and unsteady benchmark problems on unstructured grids.

Fully consistent CFD methods for incompressible flow computations

Nowadays collocated grid based CFD methods are one of the most efficient tools for computations of the flows past wind turbines. To ensure the robustness of the methods they require special attention to the well-known problem of pressure-velocity coupling. Many commercial codes to ensure the pressure-velocity coupling on collocated grids use the so-called momentum interpolation method of Rhie and Chow [1]. As known, the method and some of its widely spread modifications result in solutions, which are dependent of time step at convergence. In this paper the magnitude of the dependence is shown to contribute about 0.5% into the total error in a typical turbulent flow computation. Nevertheless if coarse grids are used, the standard interpolation methods result in much higher non-consistent behavior. To overcome the problem, a recently developed interpolation method, which is independent of time step, is used. It is shown that in comparison to other time step independent method, the method may enhance the convergence rate of the SIMPLEC algorithm up to 25 %. The method is verified using turbulent flow computations around a NACA 64618 airfoil and the roll-up of a shear layer, which may appear in wind turbine wake.

A consistent momentum interpolation method for steady and unsteady multiphase flows

A consistent momentum interpolation approach is proposed for the discretization on collocated grids of the Eulerian–Eulerian equations governing unsteady multiphase flows. The procedure, which includes a correction for interphase forces (drag), fulfills additional requirements that a multiphase method has to meet with respect to single-phase algorithms. Results reveal that the application to multiphase flows of the simple extension of the often-used formulation for single-phase flows (Choi, 1999; Shen et al., 2001) may notably result in spurious spatial oscillations in the fields, and steady solutions of transient flows that depend significantly on the time-step size. The proposed new approach, instead, is free from these deficiencies.

Fourier Analysis of the Effect of Momentum Interpolation on the SIMPLE Series Formulated on a Collocated Grid

The error amplification matrices of three SIMPLE series formulated on a staggered grid and a collocated grid using three momentum interpolation methods are derived. The effects of grid layout and momentum interpolation methods on convergence properties of SIMPLE series are checked. The results indicate that the convergence conditions are available by damping out the highest and lowest errors. The SIMPLE series show almost identical convergence rates under a wide range of useful velocity relaxation factors, regardless of grid layout and interpolation method. However, the high-frequency reduction shows significant variation with grid layout and interpolation method, which maybe referred to in multigrid application.