A robust multi‐grid pressure‐based algorithm for multi‐fluid flow at all speeds

This paper reports on the implementation and testing, within a full non‐linear multi‐grid environment, of a new pressure‐based algorithm for the prediction of multi‐fluid flow at all speeds. The algorithm is part of the mass conservation‐based algorithms (MCBA) group in which the pressure correction equation is derived from overall mass conservation. The performance of the new method is assessed by solving a series of two‐dimensional two‐fluid flow test problems varying from turbulent low Mach number to supersonic flows, and from very low to high fluid density ratios. Solutions are generated for several grid sizes using the single grid (SG), the prolongation grid (PG), and the full non‐linear multi‐grid (FMG) methods. The main outcomes of this study are: (i) a clear demonstration of the ability of the FMG method to tackle the added non‐linearity of multi‐fluid flows, which is manifested through the performance jump observed when using the non‐linear multi‐grid approach as compared to the SG and PG methods; (ii) the extension of the FMG method to predict turbulent multi‐fluid flows at all speeds. The convergence history plots and CPU‐times presented indicate that the FMG method is far more efficient than the PG method and accelerates the convergence rate over the SG method, for the problems solved and the grids used, by a factor reaching a value as high as 15. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical methods for conjunctive two‐dimensional surface and three‐dimensional sub‐surface flows

Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and sub‐surface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two‐dimensional surface flow interacting with three‐dimensional unsaturated and saturated sub‐surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two‐ and three‐dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub‐surface flow sub‐models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul'yev's methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub‐surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul'yev's downstream (S‐d) method was found to be the preferred method for two‐dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three‐dimensional sub‐surface flow model, the ADE method of Larkin (ADE‐L) and Brian's ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S‐d surface flow sub‐model give excellent results and confirm the expectation that the errors of the surface and sub‐surface sub‐models interact. The surface sub‐model dominates the accuracy and stability of the conjunctive model, whereas the sub‐surface sub‐model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub‐model. Copyright © 2000 John Wiley & Sons, Ltd.     

Three‐dimensional transient free‐surface flow of viscous fluids inside cavities of arbitrary shape

The three‐dimensional transient free‐surface flow inside cavities of arbitrary shape is examined in this study. An adaptive (Lagrangian) boundary‐element approach is proposed for the general three‐dimensional simulation of confined free‐surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free‐surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. The method is used to determine the flow field and free‐surface evolution inside cubic, rectangular and cylindrical containers. These problems illustrate the transient nature of the flow during the mixing process. Surface tension effects are also explored. Copyright © 2003 John Wiley & Sons, Ltd.     

A new multi‐domain/multi‐resolution method for large‐eddy simulation

A new multi‐domain/multi‐resolution method is presented in the framework of the large‐eddy simulation (LES). The proposed treatment at the interfaces is conceived to deal with the problem of discontinuities on the characteristic length scales met in the case of two domains having different resolutions. It gives rise to an original approach taking into account not only the discontinuous aspect on the flow fields values but also, consequently, the non‐conservative aspect of transfer of fluxes through the interfaces. This new treatment at the interfaces has been assessed successfully in the case of a subsonic compressible channel flow. Copyright © 2001 John Wiley & Sons, Ltd.     

A stress‐based least‐squares finite‐element model for incompressible Navier–Stokes equations

In this paper we present a stress‐based least‐squares finite‐element formulation for the solution of the Navier–Stokes equations governing flows of viscous incompressible fluids. Stress components are introduced as independent variables to make the system first order. Continuity equation becomes an algebraic equation and is eliminated from the system with suitable modifications. The h and p convergence are verified using the exact solution of Kovasznay flow. Steady flow past a large circular cylinder in a channel is solved to test mass conservation. Transient flow over a backward‐facing step problem is solved on several meshes. Results are compared with that obtained using vorticity‐based first‐order formulation for both benchmark problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Evaluation of Smagorinsky‐based subgrid‐scale models in a finite‐volume computation

Smagorinsky‐based models are assessed in a turbulent channel flow simulation at Re_b=2800 and Re_b=12500. The Navier–Stokes equations are solved with three different grid resolutions by using a co‐located finite‐volume method. Computations are repeated with Smagorinsky‐based subgrid‐scale models. A traditional Smagorinsky model is implemented with a van Driest damping function. A dynamic model assumes a similarity of the subgrid and the subtest Reynolds stresses and an explicit filtering operation is required. A top‐hat test filter is implemented with a trapezoidal and a Simpson rule. At the low Reynolds number computation none of the tested models improves the results at any grid level compared to the calculations with no model. The effect of the subgrid‐scale model is reduced as the grid is refined. The numerical implementation of the test filter influences on the result. At the higher Reynolds number the subgrid‐scale models stabilize the computation. An analysis of an accurately resolved flow field reveals that the discretization error overwhelms the subgrid term at Re_b=2800 in the most part of the computational domain. Copyright © 2002 John Wiley & Sons, Ltd.     

The multi‐stage centred‐scheme approach applied to a drift‐flux two‐phase flow model

For two‐phase flow models, upwind schemes are most often difficult do derive, and expensive to use. Centred schemes, on the other hand, are simple, but more dissipative. The recently proposed multi‐stage (MUSTA ) method is aimed at coming close to the accuracy of upwind schemes while retaining the simplicity of centred schemes. So far, the MUSTA approach has been shown to work well for the Euler equations of inviscid, compressible single‐phase flow. In this work, we explore the MUSTA scheme for a more complex system of equations: the drift‐flux model, which describes one‐dimensional two‐phase flow where the motions of the phases are strongly coupled. As the number of stages is increased, the results of the MUSTA scheme approach those of the Roe method. The good results of the MUSTA scheme are dependent on the use of a large‐enough local grid. Hence, the main benefit of the MUSTA scheme is its simplicity, rather than CPU ‐time savings. Copyright © 2006 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.     

Transient two‐layer thin‐film flow

The transient two‐layer thin‐film planar flow is investigated theoretically in this study. The interplay among inertia, viscous and surface/interfacial tension is emphasized. It is found that the film and interface profiles, as well as the flow field, are strongly influenced by the viscosity ratio, velocity and film thickness ratios at inception, and the surface‐to‐interfacial tension ratio. The nonlinear stability of the steady state reveals the formation of a solitary wave after flow inception, which propagates in the form of a convective instability, with the steady state recovered only in the tail (upstream) region of the wave. In the presence of surface/interfacial tension, surface modulation appears, which grows in wavelength and amplitude with position. The flow is found to be particularly stable for higher viscosity of the lower film layer. Copyright © 2010 John Wiley & Sons, Ltd.     

Pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist

We propose a pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist. Unlike the original thermo–Cubic Interpolated Propagation Combined Unified Procedure (CIP‐CUP) method proposed by Himeno et al (Transactions of the Japan Society of Mechanical Engineers, Series B, 2003), we split the advection term of the governing equations into a conservation part and into the rest. The splitting of advection term has two advantages. One is the high degree of freedom in choosing discretization schemes such as central‐difference schemes, upwind schemes, and Total Variation Diminishing (TVD) schemes. The other is the ease of implementation on unstructured grids. The advantages enable the analyses of various flows such as turbulent and supersonic ones in actual complicated boundaries. Therefore, the solver is useful for practical analyses. The solver was validated on the following test cases: subsonic single‐phase flows, incompressible single‐phase turbulent flows, and incompressible gas‐liquid two‐phase flows. With unstructured grids, we obtained the equivalent results as the ones with structured grids. After the validations, subsonic jet impinging on a water pool was calculated and compared with experimental results. It was confirmed that the calculated results were consistent with the experimental ones.     

Incompressible multiphase flow and encapsulation simulations using the moment‐of‐fluid method

A moment‐of‐fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment‐of‐fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi‐Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block‐structured adaptive grid. The new method is applied to multiphase problems illustrating contact‐line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two‐dimensional, three‐dimensional axisymmetric (R–Z), and three‐dimensional (X–Y–Z) coordinate systems. Copyright © 2015 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.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

A weakly compressible MPS method for modeling of open‐boundary free‐surface flow

A mesh‐free particle method, based on the moving particle semi‐implicit (MPS) interaction model, has been developed for the simulation of two‐dimensional open‐boundary free‐surface flows. The incompressibility model in the original MPS has been replaced with a weakly incompressible model. The effect of this replacement on the efficiency and accuracy of the model has been investigated. The new inflow–outflow boundary conditions along with the particle recycling strategy proposed in this study extend the application of the model to open‐boundary problems. The final model is able to simulate open‐boundary free surface flow in cases of large deformation and fragmentation of free surface. The models and proposed algorithms have been validated and applied to sample problems. The results confirm the model's efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

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 subdomain boundary element method for high‐Reynolds laminar flow using stream function‐vorticity formulation

The paper presents a new formulation of the integral boundary element method (BEM) using subdomain technique. A continuous approximation of the function and the function derivative in the direction normal to the boundary element (further ‘normal flux’) is introduced for solving the general form of a parabolic diffusion‐convective equation. Double nodes for normal flux approximation are used. The gradient continuity is required at the interior subdomain corners where compatibility and equilibrium interface conditions are prescribed. The obtained system matrix with more equations than unknowns is solved using the fast iterative linear least squares based solver. The robustness and stability of the developed formulation is shown on the cases of a backward‐facing step flow and a square‐driven cavity flow up to the Reynolds number value 50 000. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.