Lagrangian slug flow modeling and sensitivity on hydrodynamic slug initiation methods in a severe slugging case

Severe slugging is a dynamic two-phase flow phenomenon with regular liquid accumulation and blow-out in flow-line riser geometries. This paper discusses the applicability of a slug tracking model on a case where hydrodynamic slug initiation in a horizontal part of the pipeline upstream the riser base affects the severe slugging cycle period. The given experimental case is from the Shell laboratories in Amsterdam: air–water flow in a 100 m long pipe (65 m horizontal and 35 m −2.54° downwards) followed by a 15 m long vertical riser.A Lagrangian slug and bubble tracking model is described. A two-fluid model is applied in the bubble region and the slug region is treated as incompressible flow, with an integral momentum equation. Slug initiation from unstable stratified flow can be captured directly by solving the two-fluid model on a fine grid (a hybrid capturing and tracking scheme). Alternatively, slug initiation can be made from sub grid models, allowing for larger grid sizes. The sub grid models are based on the two established flow regime transition criteria derived from the stability of stratified flow and from the limiting solution of the unit cell slug flow model.Sensitivity studies on hydrodynamic slug initiation models on the severe slugging characteristics are presented. No hydrodynamic slug initiation (e.g. large grid size in the capturing scheme) overestimates the severe slug period compared with the experiments. Slug capturing and sub grid initiation models both give good predictions for small grid sizes (provided the detailed inlet configuration is included in the capturing case). Good predictions are also shown for larger grid sizes (factor of 50) and sub grid initiation models.The numerical tests show that correct prediction of the severe slugging cycle is sensitive to the initiation of upstream hydrodynamic slugs, but less sensitive to the local structure of the slug flow (frequencies and lengths) in the upstream region.     

Extension of a high‐resolution scheme to 1D liquid–gas flow

This paper presents the extension of a high‐resolution conservative scheme to the one‐dimensional one‐pressure six‐equation two‐fluid flow model. Only mixtures of water and air have been considered in this study, both fluids have been characterized using simple equations of state, namely stiffened gas for the liquid phase and perfect gas for the gas phase. The resulting scheme is explicit and first‐order accurate in space and time. A second‐order version of the scheme has also been derived using the MUSCL strategy and slope limiters. Some numerical results show the good capabilities of this type of schemes in the solution of discontinuities in two‐fluid flow problems, all of them are based on water/air numerical benchmarks widely used in the two‐phase flow literature. Copyright © 2005 John Wiley & Sons, Ltd.     

A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations

In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy.     

Shock‐capturing with natural high‐frequency oscillations

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock‐capturing under the influence of natural high‐frequency oscillations. The conjugate low‐ and high‐pass filters are constructed based on the principle of the discrete singular convolution (DSC), a local spectral method. The accuracy and resolution of the DSC basic algorithm are accessed with a one‐dimensional advection equation. Two Euler systems, the advection of an isotropic vortex flow and the interaction of shock–entropy wave are utilized to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock–entropy wave interaction. Numerical experiments indicate that the CFOR scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical study of wave propagation in compressible two‐phase flow

We propose a new model and a solution method for two‐phase two‐fluid compressible flows. The model involves six equations obtained from conservation principles applied to a one‐dimensional flow of gas and liquid mixture completed by additional closure governing equations. The model is valid for pure fluids as well as for fluid mixtures. The system of partial differential equations with source terms is hyperbolic and has conservative form. Hyperbolicity is obtained using the principles of extended thermodynamics. Features of the model include the existence of real eigenvalues and a complete set of independent eigenvectors. Its numerical solution poses several difficulties. The model possesses a large number of acoustic and convective waves and it is not easy to upwind all of these accurately and simply. In this paper we use relatively modern shock‐capturing methods of a centred‐type such as the total variation diminishing (TVD) slope limiter centre (SLIC) scheme which solve these problems in a simple way and with good accuracy. Several numerical test problems are displayed in order to highlight the efficiency of the study we propose. The scheme provides reliable results, is able to compute strong shock waves and deals with complex equations of state. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐dimensional modeling for stability analysis of two‐phase stratified flow

The effect of wavelength and relative velocity on the disturbed interface of two‐phase stratified regime is modeled and discussed. To analyze the stability, a small perturbation is imposed on the interface. Growth or decline of the disturbed wave, relative velocity, and surface tension with respect to time will be discussed numerically. Newly developed scheme applied to a two‐dimensional flow field and the governing Navier–Stokes equations in laminar regime are solved. Finite volume method together with non‐staggered curvilinear grid is a very effective approach to capture interface shape with time. Because of the interface shape, for any time advancement, a new grid is performed separately on each stratified field, liquid, and gas regime. The results are compared with the analytical characteristics method and one‐dimensional modeling. This comparison shows that solving the momentum equation including viscosity term leads to physically more realistic results. In addition, the newly developed method is capable of predicting two‐phase stratified flow behavior more precisely than one‐dimensional modeling. It was perceived that the surface tension has an inevitable role in dissipation of interface instability and convergence of the two‐phase flow model. Copyright © 2009 John Wiley & Sons, Ltd.     

Depth‐integrated,non‐hydrostatic model with grid nesting for tsunami generation,propagation, and run‐up

Tsunamis generated by earthquakes involve physical processes of different temporal and spatial scales that extend across the ocean to the shore. This paper presents a shock‐capturing dispersive wave model in the spherical coordinate system for basin‐wide evolution and coastal run‐up of tsunamis and discusses the implementation of a two‐way grid‐nesting scheme to describe the wave dynamics at resolution compatible to the physical processes. The depth‐integrated model describes dispersive waves through the non‐hydrostatic pressure and vertical velocity, which also account for tsunami generation from dynamic seafloor deformation. The semi‐implicit, finite difference model captures flow discontinuities associated with bores or hydraulic jumps through the momentum‐conserved advection scheme with an upwind flux approximation. The two‐way grid‐nesting scheme utilizes the Dirichlet condition of the non‐hydrostatic pressure and both the horizontal velocity and surface elevation at the inter‐grid boundary to ensure propagation of dispersive waves and discontinuities across computational grids of different resolution. The inter‐grid boundary can adapt to bathymetric features to model nearshore wave transformation processes at optimal resolution and computational efficiency. A coordinate transformation enables application of the model to small geographic regions or laboratory experiments with a Cartesian grid. A depth‐dependent Gaussian function smoothes localized bottom features in relation to the water depth while retaining the bathymetry important for modeling of tsunami transformation and run‐up. Numerical experiments of solitary wave propagation and N‐wave run‐up verify the implementation of the grid‐nesting scheme. The 2009 Samoa Tsunami provides a case study to confirm the validity and effectiveness of the modeling approach for tsunami research and impact assessment. Copyright © 2010 John Wiley & Sons, Ltd.     

A new fully non‐hydrostatic 3D free surface flow model for water wave motions

A new fully non‐hydrostatic model is presented by simulating three‐dimensional free surface flow on a vertical boundary‐fitted coordinate system. A projection method, known as pressure correction technique, is employed to solve the incompressible Euler equations. A new grid arrangement is proposed under a horizontal Cartesian grid framework and vertical boundary‐fitted coordinate system. The resulting model is relatively simple. Moreover, the discretized Poisson equation for pressure correction is symmetric and positive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method. Several test cases of surface wave motion are used to demonstrate the capabilities and numerical stability of the model. Comparisons between numerical results and analytical or experimental data are presented. It is shown that the proposed model could accurately and effectively resolve the motion of short waves with only two layers, where wave shoaling, nonlinearity, dispersion, refraction, and diffraction phenomena occur. Copyright © 2010 John Wiley & Sons, Ltd.     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallel three‐dimensional simulation of the injection molding process

This paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 John Wiley & Sons, Ltd.     

A high‐resolution method for compressible two‐fluid flows and simulation of three‐dimensional shock–bubble interactions

A high‐resolution method is developed to capture the material interfaces of compressible two‐fluid flows in multiple dimensions. A fluid mixture model system with single velocity and pressure is used, and viscous effect can also be taken into account. A consistent thermodynamic law based on the assumption of pressure equilibrium is employed to describe the thermodynamic behaviors of the pure fluids and mixture of two components. The splitting and unsplit Eulerian formulations of piecewise parabolic method are extended to numerically integrate the hyperbolic part of the model system, whereas the system of diffusion equations is solved using an explicit, central difference scheme. The block‐structured adaptive mesh refinement (AMR) capability is built in the hydrodynamic code to locally improve grid resolution. The resulting method is verified to be at least second‐order accurate in space. Numerical results show that the discontinuities, particularly contact discontinuities, can be resolved sharply. The use of AMR allows flow features at disparate scales to be resolved sufficiently. In addition, three‐dimensional shock–bubble interactions are simulated to investigate effects of Mach number on bubble evolution. The flow structures including those peculiar to three‐dimensional bubble are resolved correctly, and some physical phenomena with increasing Mach number are reported. Copyright © 2012 John Wiley & Sons, Ltd.     

Development of a generalized multi‐layer model for 3‐D simulation of free surface flows

A mathematical model was developed for three‐dimensional (3‐D) simulation of free surface flows. In this model, the flow depth is divided into a number of layers and shallow water equations are integrated in each layer to derive the hydrodynamic equations. To give a general form to this model, each layer is assumed to be non‐horizontal with varying thickness in the flow domain. A non‐orthogonal curvilinear coordinate system is employed in the model, to allow for flexibility in dealing with the irregular geometry of natural watercourses. Due to the similarity in governing equations, two‐dimensional (2‐D) depth averaged programs can be developed into a multi‐layer model. The development for a depth averaged program and its numerical scheme is described in this paper. Experimental data and semi‐analytical solutions are used to evaluate the performance of the model. Three different cases of open channel flow are tested: 1‐flow in a straight open channel, 2‐the flow development region in a channel, and 3‐flow in a meandering channel. It is shown that the model has the capability to predict velocity distribution and secondary flows in complex 3‐D flow conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Development of a compressive surface capturing formulation for modelling free‐surface flow by using the volume‐of‐fluid approach

With the aim of accurately modelling free‐surface flow of two immiscible fluids, this study presents the development of a new volume‐of‐fluid free‐surface capturing formulation. By building on existing volume‐of‐fluid approaches, the new formulation combines a blended higher resolution scheme with the addition of an artificial compressive term to the volume‐of‐fluid equation. This reduces the numerical smearing of the interface associated with explicit higher resolution schemes while limiting the contribution of the artificial compressive term to ensure the integrity of the interface shape is maintained. Furthermore, the computational efficiency of the the higher resolution scheme is improved through the reformulation of the normalised variable approach and the implementation of a new higher resolution blending function. The volume‐of‐fluid equation is discretised via an unstructured vertex‐centred finite volume method and solved via a Jacobian‐type dual time‐stepping approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Computation of free‐surface flows with local mesh adaptation

This paper describes a modern free‐surface capturing strategy implemented in an unstructured finite‐volume viscous flow solver that can handle moving grids composed of arbitrary‐shaped control volumes. An adaptive mesh strategy is fully integrated in the code making it a single tool for dynamically maintaining a prescribed density of grid points around the steady or unsteady interface between air and water. The whole adaptive procedure is described in detail. The efficiency of the overall approach is examined on two‐ and three‐dimensional hydrodynamic applications. The adaptive strategy achieves interesting gains in terms of computational and human efforts compared to single‐mesh computations. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

This paper describes a numerical solver of well‐balanced, 2D depth‐averaged shallow water‐sediment equations. The equations permit variable horizontal fluid density and are designed to model water‐sediment flow over a mobile bed. A Godunov‐type, Harten–Lax–van Leer contact (HLLC) finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws that describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi‐analytical solutions for bedload transport and suspended sediment transport, respectively. The well‐balanced property of the equations is verified for a variable‐density dam break flow over discontinuous bathymetry. Simulations of an idealised dam‐break flow over an erodible bed are in excellent agreement with previously published results, validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable‐density governing equations. Flow hydrodynamics and final bed topography of a laboratory‐based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water‐sediment models to the choice of closure relationships. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.