Hydrodynamical instability initiation in two-phase stratified flow using Spectral Method

In this article, the hydrodynamical instability initiation criterion in two-phase stratified flow in a horizontal duct is examined. The nonlinear two mass and two momentum conservation equations are used for numerical simulation using the two-phase two-fluid model. The model is solved using the Finite Volume and Spectral Methods, respectively. This paper is the first to utilize the Spectral Method for the simulation of two-phase flow problems. Using the Spectral Method, we show that the numerical error and CPU time decreases noticeably relative to the Finite Volume Method. The well established Kelvin–Helmholtz (K–H) instability is selected for the test case and comparison. The results taken from each set of computer codes developed in this paper are highly compatible with the theoretical and experimental results of previous researchers who used alternative numerical methods. The results obtained from the Spectral Method in comparison with the results of other well known codes exhibit greater consistency with prior analytical results, but with much smaller computer calculation time. The step taken in the present study shows a positive progress in two-phase two-fluid model numerical solution with hydrostatic assumption. It is recommended the research to be continued with two-phase two-fluid model but with hydrodynamical assumption.     

A positive splitting method for mixed hyperbolic‐parabolic systems

In this article we present a method of lines approach to the numerical solution of a system of coupled hyperbolic—parabolic partial differential equations (PDEs). Special attention is paid to preserving the positivity of the solution of the PDEs when this solution is approximated numerically. This is achieved by using a flux‐limited spatial discretization for the hyperbolic equation. We use splitting techniques for the solution of the resulting large system of stiff ordinary differential equations. The performance of the approach applied to a biomathematical model is compared with the performance of standard methods. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 152–168, 2001     

A modified sensitivity equation method for the Euler equations in presence of shocks

The continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti-diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 1: analysis

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations. The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Transient modeling of non-isothermal, dispersed two-phase flow in natural gas pipelines

Unsteady-state or transient two-phase flow, caused by any change in rates, pressures or temperature at any location in a two-phase flow line, may last from a few seconds to several hours. In general, these changes are an order of magnitude longer than the transient encountered during single-phase flow. The primary reason for this phenomenon is that the velocity of wave propagation in a two-phase mixture is significantly slower. Interfacial transfer of mass, momentum and energy further complicate the problem. It is primarily due to the numerical difficulties anticipated in accurately modeling transient two-phase flow that the state of the art in this important area is restricted to a handful of studies with direct applicability to petroleum and gas engineering. A limited amount of information on the subject of two-phase transport phenomena is available in the petroleum engineering literature. Most of the publications for two-phase flow of gas assume that temperature is constant over the entire length of the pipeline.This study is the first effort to simulate the non-isothermal, one-dimensional, transient homogenous two-phase flow gas pipeline system using two-fluid conservation equations. The modified Peng–Robinson equation of state is used to calculate the vapor–liquid equilibrium in multi-component natural gas to find the vapor and liquid compressibility factors. Mass transfer between the gas and the liquid phases is treated rigorously through flash calculation, making the algorithm capable of handling retrograde condensation. The liquid droplets are assumed to be spheres of uniform size, evenly dispersed throughout the gas phase.The method of solution is the fully implicit finite difference method. This method is stable for gas pipeline simulations when using a large time step and therefore minimizes the computation time. The algorithm used to solve the non-linear finite difference thermo-fluid equations for two-phase flow through a pipe is based on the Newton–Raphson method.The results show that the liquid condensate holdup is a strong function of temperature, pressure, mass flow rate, and mixture composition. Also, the fully implicit method has advantages, such as the guaranteed stability for large time step, which is very useful for simulating long-term transients in natural gas pipeline systems.     

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.     

Optimal Control for Traffic Flow Networks

We consider traffic flow models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented.This author was supported by Deutsche Forschungsgemeinschaft (DFG), Grant KL 1105/5.     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

An Algorithm for Minimum L-Infinity Solution of Under-determined Linear Systems

This paper presents a primal method for finding the minimum L-infinity solution to under-determined linear systems of equations. The method is a two-phase method. Line search is performed at both phases. We establish a condition for a direction to be descent. The convergence proof of the method is shown. Expedient numerical schemes can be used whenever appropriate. Results are presented, which show the superiority of the method over some well-known methods.     

An approximation of stochastic hyperbolic equations: case with Wiener process

In the present paper, the two‐step difference scheme for the Cauchy problem for the stochastic hyperbolic equation is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of four problems for hyperbolic equations are obtained. The theoretical statements for the solution of this difference scheme are supported by the results of the numerical experiment. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element analysis of flood discharge atomization based on water–air two-phase flow

Flood discharge atomization is a phenomenon of water fog diffusion caused by the discharge of water from a spillway structure, which brings strong wind and heavy rainfall. These unnatural winds and rainfall are harmful for the safe operation of hydropower stations with high water heads. Compared to the method of prototype observations, physical models and mathematical models, which are semi-theoretical and semi-empirical, numerical simulation methods have the advantage of being not limited by a similar scale and are more economical. A finite element model is presented to simulate flood discharge atomization based on water–air two-phase flow in this paper. Equations governing flood discharge atomization are composed of partial differential equations of mass and momentum conservation laws with unknowns for pressure, velocity and the water concentration. The finite element method is used to solve the governing equations by adopting appropriate solution strategies to increase the convergence and numerical stability. Then, the finite element model is applied to a practical project, the Shuibuya hydropower station, which experienced a flood discharge in 2016. Simulation results show that the proposed model can simulate flood discharge atomization with efficient convergence and numerical stability in three dimensions, and good agreement was observed between numerical simulations and prototype observational data. Based on the simulation results, the mechanism of flood discharge atomization was analyzed.     

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

Solution of two-phase flow problems in porous media via an alternating-direction finite element method

The application of an alternating-direction finite element solution procedure to two-phase immiscible displacement problems in porous media is illustrated. This solution scheme provides for rapid solution of the discrete problem, due to the narrow banded matrices involved, with an accuracy which is comparable to that of standard finite element approximations. The governing partial differential equations for immiscible two-phase porous media flow are given and their discretization, via a Laplace-modified time stepping scheme, is presented. Iterative improvement of the time stepping scheme is also considered and numerical examples are provided which demonstrate the saving in computational time which can be achieved.     

Piecewise parabolic method on local stencil for gasdynamic simulations

A numerical method based on piecewise parabolic difference approximations is proposed for solving hyperbolic systems of equations. The design of its numerical scheme is based on the conservation of Riemann invariants along the characteristic curves of a system of equations, which makes it possible to discard the four-point interpolation procedure used in the standard piecewise parabolic method (PPM) and to use the data from the previous time level in the reconstruction of the solution inside difference cells. As a result, discontinuous solutions can be accurately represented without adding excessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new method is compared with PPM by solving test problems for the linear advection equation and the inviscid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. A technique for solving the gas dynamics equations is described and tested for several one-and two-dimensional problems.     

Numerical investigation of the effect of helix angle and leaf margin on the flow pattern and the performance of the axial flow cyclone separator

A two-stage turbulence model based on the RNG κ–ε model combined with the Reynolds stress model is developed in this paper to analyze the gas flow in an axial flow cyclone separator. Five representative simulation cases are obtained by changing the helix angle and leaf margins of the cyclone. The pressure field and velocity field of the five cases are simulated, and then the effects of helix angle and leaf margins on the internal flow field of the cyclone are analyzed. When the continuum fluid (air) flow is relatively convergent, the discrete particle phase is added into the continuous phase and the gas-solid two-phase flow is simulated. One-way coupling method is used to solve the two-phase flow and a stochastic trajectory model is implemented for simulation of the particle phase. Finally, the pressure drop and separation efficiency of one case are measured and compare quantitatively well with the numerical results, which validates the reliability and accuracy of the simulation method based on the two-stage turbulence model.     

Solving hyperbolic conservation laws using multiquadric quasi‐interpolation

In this article, we apply the univariate multiquadric (MQ) quasi‐interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi‐interpolation corresponding to periodic and inflow‐outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi‐interpolation to approximate the spatial derivative of the differential equation and a low‐order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one‐dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is ??(τ), where τ is the temporal step. We can improve the accuracy by using the high‐order quasi‐interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

HIGH RESOLUTION SCB SCHEME FOR HYPERBOLIC SYSTEMS OF 2-D CONSERVATION LAWS

1.IntroductionConsidernumericalsolutionsoftheinitialvalueproblemforhyperbolicconservationlawsinonedimensionwhereu~(.ul,u2,...,"m)"andf(u)~(fi(u),fZ(u),',fm(u))".Andconservationlawsintwodimensionswhereu~(ul,u25...,"m)',f(u)~(fi(u),fZ(u),'?fm(u))',andg(u)~(gi(u),gZ(M),',gm(U))".FOrthescalarconservationlawsinonedimension,theTVDconceptbyA.Ha.te.[3]iswidelyacceptedtodesignthenumericalschemesfortheoreticalpurposesandpracticalapplications.Thetotalvariationofagridfunction{uj}denotedbyTV(u)…