On the steady solution of non-linear advection equations in steady recirculating flows

Many physical problems like short fiber suspensions flows or viscoelastic flows are modeled by linear and non-linear advection equations. Many of the experimental and industrial flows show often steady recirculating areas which introduce some additional difficulties in the numerical simulation. Actually, the advection equation is supposed to have a steady solution in these steady recirculating flows but neither boundary conditions nor initial conditions are known in such flows. In this paper, we present accurate techniques to solve non-linear advection equations defined in steady recirculating flows. These techniques combine a standard treatment of the non-linearity with a more original treatment of the associated linear problems.     

Digital marbling: a multiscale fluid model

This paper presents a multiscale fluid model based on mesoscale dynamics and viscous fluid equations as a generic tool for digital marbling purposes. The model uses an averaging technique on the adaptation of a stochastic mesoscale model to obtain the effect of fluctuations at different levels. It allows various user controls to simulate complex flow behaviors as in traditional marbling techniques, as well as laminar and turbulent flows. Material transport is based on an improved advection solution to be able to match the highly detailed, sharp fluid interfaces in marbling patterns. In the transport model, two reaction models are introduced to create different effects and to simulate density fluctuations.     

A discrete Boltzmann equation model for two-phase shallow granular flows

In this paper a Discrete Boltzmann Equation model (hereinafter DBE) is proposed as solution method of the two-phase shallow granular flow equations, a complex nonlinear partial differential system, resulting from the depth-averaging procedure of mass and momentum equations of granular flows. The latter, as e.g. a debris flow, are flows of mixtures of solid particles dispersed in an ambient fluid.The reason to use a DBE, instead of a more conventional numerical model (e.g. based on Riemann solvers), is that the DBE is a set of linear advection equations, which replaces the original complex nonlinear partial differential system, while preserving the features of its solutions. The interphase drag function, an essential characteristic of any two-phase model, is accounted for easily in the DBE by adding a physically based term. In order to show the validity of the proposed approach, the following relevant benchmark tests have been considered: the 1D simple Riemann problem, the dam break problem with the wet–dry transition of the liquid phase, the dry bed generation and the perturbation of a state at rest in 2D. Results are satisfactory and show how the DBE is able to reproduce the dynamics of the two-phase shallow granular flow.     

Numerical simulation of Richtmyer–Meshkov instability driven by imploding shocks

In this paper, the classical piecewise parabolic method (PPM) is generalized to compressible two-fluid flows, and is applied to simulate Richtmyer–Meshkov instability (RMI) induced by imploding shocks. We use the compressible Euler equations together with an advection equation for volume fraction of one fluid component as model system, which is valid for both pure fluid and two-component mixture. The Lagrangian-remapping version of PPM is employed to solve the governing equations with dimensional-splitting technique incorporated for multi-dimensional implementation, and the scheme proves to be non-oscillatory near material interfaces. We simulate RMI driven by imploding shocks, examining cases of single-mode and random-mode perturbations on the interfaces and comparing results of this instability in planar and cylindrical geometries. Effects of perturbation amplitude and shock strength are also studied.     

改进流体体积法及重建光滑三维液面

通过改进流体体积法(VOF)方程求解部分，提高了离散模型中表面单元和液体内部对流的精度，从而在整体上减小了对流产生的体积误差；为逼真地展现具有复杂表面的运动液体场景，提出一种基于几何搜索的表面重建方法——种子一绕空方法．该方法可以将VOF方法生成的带缝隙表面用C^0连接的三角面片集表示，以获得光滑连续的液面表示，最后通过实例实现了较低密度网格下光滑地表现复杂液面。     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Finite element solution for advection and natural convection flows

A new equal order velocity—pressure finite element procedure is presented for the calculation of 2-D viscous, incompressible flows of a recirculating nature. As in the finite difference procedures, velocity and pressure e uncoupled and the equations are solved one after the other. In this splitting-up method, an auxilary velocity field is computed first, which accounts for all contributions to the acceleration, except pressure, and satisfies the velocity boundary conditions. Then, the final velocities are evaluated by adding to the auxilary velocities pressure contributions which are computed to satisfy the continuity equation. The effectiveness is illustrated via example problems of 2-D advection and natural convection flows.     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces

We provide an overview of the finite element methods we developed for fluid dynamics problems. We focus on stabilized formulations and moving boundaries and interfaces. The stabilized formulations are the streamline-upwind/Petrov-Galerkin (SUPG) formulations for compressible and incompressible flows and the pressure-stabilizing/Petrov-Galerkin (PSPG) formulation for incompressible flows. These are supplemented with the discontinuity-capturing directional dissipation (DCDD) for incompressible flows and the shock-capturing terms for compressible flows. Determination of the stabilization and shock-capturing parameters used in these formulations is highlighted. Moving boundaries and interfaces include free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed for this class of problems can be classified into two main categories: interface-tracking and interface-capturing techniques. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space-time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques were developed for two-fluid flows. They are based on the stabilized formulation, over typically non-moving meshes, of both the flow equations and an advection equation. The advection equation governs the time-evolution of an interface function marking the interface location. We also describe some of the additional methods and ideas we introduced to increase the scope and accuracy of these two classes of techniques. Among them is the enhanced-discretization interface-capturing technique (EDICT), which was developed to increase the accuracy in capturing the interface. Also among them is the mixed interface-tracking/interface-capturing technique (MITICT), which was introduced for problems that involve both interfaces that can be accurately tracked with a moving-mesh method and interfaces that call for an interface-capturing technique.     

A collocation method based on the influence matrix technique for Navier-Stokes problems in annular domains

A spectral collocation method is proposed for the solution of the time-dependent Navier-Stokes and energy equations of a Boussinesq fluid inside an annular cavity. The time integration is based on the Adams-Bashforth scheme and on the second order backward differentiation formula. The influence matrix technique results in the resolution of Helmholtz and Poisson equations with Dirichlet boundary conditions. The solutions are validated with respect to former spectral Tau-Chebyshev solutions. Preliminary results concern the simulation of axisymmetric flows submitted to the buoyancy force, to the rotation and to source-sink fluxes.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.     

Solution properties and approximate Riemann solvers for two-layer shallow flow models

The 4 × 4 system of governing equations for two-layer shallow flow models is known to exhibit particular behaviours such as loss of hyperbolicity under certain flow configurations. An eigenvalue analysis of the conservation part of the equations shows that the loss of hyperbolicity is due only to the reaction exerted by each fluid onto the other at the interface between the fluids. Three Riemann solvers derived from the HLL formalism are presented. In the first solver, the pressure-induced terms are accounted for by the source term; in the second solver, they are incorporated into the fluxes; the third solver uses the same formulation as the first, except that the mass and momentum balance for the bottom layer are replaced with the balance equations for the system formed by the two layers as a whole. Numerical results using the three solvers are presented for (1) static conditions such as two fluids of identical densities at rest above each other, (2) dam-break flows involving the collapse of a body of light fluid over a uniform layer of a denser fluid, and (3) Liska and Wendroff’s ill-posed test cases [24] involving two-layer flows over a topographic bump. The three solvers produce quasi-undistinguishable results for the dam-break flows, and produce sharp solutions over the full range of density ratio, from 0 to 1. However, only the third solver allows a strict preservation of static configurations. Moreover, a method is proposed to assess the convergence of the numerical solutions in the configurations for which no analytical solution can be obtained.     

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.     

A calculation procedure for three-dimensional steady recirculating flows using multigrid methods

An efficient finite difference calculation procedure for three-dimensional recirculating flows is presented. The algorithm is based on a coupled solution of the three-dimensional momentum and continuity equations in primitive variables by the multigrid technique. A symmetrical coupled Gauss-Seidel technique is used for iterations and is observed to provide good rates of smoothing. Calculations have been made of the fluid motion in a three-dimensional cubic cavity with a moving top wall. The efficiency of the method is demonstrated by performing calculations at different Reynolds numbers with finite difference grids as large as 66 × 66 × 66 nodes. The CPU times and storage requirements for these calculations are observed to be very modest. The algorithm has the potential to be the basis for an efficient general-purpose calculation procedure for practical fluid flows.     

Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces

We propose a fast, explicit numerical method for computing approximations for the immersed boundary problem in which the boundaries that separate the fluid into two regions are stiff. In the numerical computations of such problems, one frequently has to contend with numerical instability, as the stiff immersed boundaries exert large forces on the local fluid. When the boundary forces are treated explicitly, prohibitively small time-steps may be required to maintain numerical stability. On the other hand, when the boundary forces are treated implicitly, the restriction on the time-step size is reduced, but the solution of a large system of coupled non-linear equations may be required. In this work, we develop an efficient method that combines an integral equation approach with the immersed interface method. The present method treats the boundary forces explicitly. To reduce computational costs, the method uses an operator-splitting approach: large time-steps are used to update the non-stiff advection terms, and smaller substeps are used to advance the stiff boundary. At each substep, an integral equation is computed to yield fluid velocity local to the boundary; those velocity values are then used to update the boundary configuration. Fluid variables are computed over the entire domain, using the immersed interface method, only at the end of the large advection time-steps. Numerical results suggest that the present method compares favorably with an implementation of the immersed interface method that employs an explicit time-stepping and no fractional stepping.     

Analytical Investigation of the Dynamics of a Hydraulic Fracture Using the Principle of Incomplete Coupling

An investigation of a self-similar solution of a coupled problem on the creeping flows of a viscous fluid in a hydraulic fracture and the strain and flow in the external poroelastic medium induced by them. The process is governed by injection fluid into a well. The flow in the fracture is described by the Stokes hydrodynamic equations in the approximation of the lubricant layer. The external problem is described by the equations of poroelasticity. The variant of a uniform pressure in the fracture is considered for three-dimensional and two-dimensional cases. In the second case, a self-similar solution can be obtained in an analytical form.     

A phase-field method for interface-tracking simulation of two-phase flows

For interface-tracking simulation of two-phase flows with a high density ratio, we propose a computational method, NS–PFM, combining Navier–Stokes (NS) equations with phase-field model (PFM) based on the free energy theory. Through the numerical simulations, it was confirmed that (1) the volume flux derived from chemical potential gradient in the Cahn–Hilliard equation of PFM plays an important role in advection and reconstruction of interface, and (2) the NS–PFM gives good predictions for the motions of immiscible, incompressible, isothermal two-phase fluid, such as air–water system, without using conventional interface-tracking techniques.